- Count the rows and columns rather than every square one at a time. - What multiplication fact matches this arrangement? - The two numbers multiplied are a factor pair of the total.

1. The array has \(3\) rows and \(4\) columns. 2. Since \(3 \times 4 = 12\), \(3\) and \(4\) are a factor pair of \(12\).

\(12\) squares; factor pair \((3, 4)\).

- Could each pair describe the rows and columns of a rectangle? - Imagine twelve squares arranged in equal rows.

1. A factor pair multiplies to the number. 2. \(2 \times 6 = 12\), but \(5 \times 7 = 35\), so only \((2,6)\) is a factor pair of \(12\).

\((2,6)\), because \(2 \times 6 = 12\).

- What does multiplying a number by one do? - Start with the simplest possible positive factor. - Write the multiplication fact that confirms your pair.

1. One times any whole number gives that number. 2. Therefore \(1 \times 19 = 19\), so \((1,19)\) is a factor pair.

\((1,19)\), because \(1 \times 19 = 19\).

- Think of a multiplication fact in the four-times table. - What number of groups of four makes the target? - Check by multiplying both numbers in your pair.

1. Find the missing factor using \(28 \div 4 = 7\). 2. The multiplication fact is \(4 \times 7 = 28\), giving factor pair \((4,7)\).

\(7\); factor pair \((4,7)\).

- Build each factor list from factor pairs so that no factor is missed. - For part c), exclude only pairs that use \(1\). - For part d), compare the completed factor lists rather than the sizes of the numbers.

1. The factor pairs of \(56\) are \(1\times56\), \(2\times28\), \(4\times14\), and \(7\times8\). Its positive factors are \(\{1,2,4,7,8,14,28,56\}\). 2. The factor pairs of \(81\) are \(1\times81\), \(3\times27\), and \(9\times9\). Its positive factors are \(\{1,3,9,27,81\}\). 3. The factor pairs of \(56\) with both factors greater than \(1\) are \((2,28)\), \((4,14)\), and \((7,8)\). 4. The number \(56\) has \(8\) positive factors and \(81\) has \(5\), so \(56\) has \(8-5=3\) more positive factors.

a) \(\{1,2,4,7,8,14,28,56\}\) b) \(\{1,3,9,27,81\}\) c) \((2,28)\), \((4,14)\), \((7,8)\) d) \(56\) has \(3\) more positive factors than \(81\).

- Check several multiplication facts to see where each product appears. - Look for factor pairs that are not just reversals of one another. - Can doubling one factor and halving the other produce another whole-number pair?

1. List factor pairs whose product is each number, without counting a reversed pair twice. 2. For \(24\), two pairs are \(3 \times 8\) and \(4 \times 6\). 3. For \(12\), two pairs are \(2 \times 6\) and \(3 \times 4\). 4. For \(20\), two pairs are \(2 \times 10\) and \(4 \times 5\). 5. For \(30\), two pairs are \(3 \times 10\) and \(5 \times 6\).

a) \(3 \times 8\) and \(4 \times 6\) b) \(2 \times 6\) and \(3 \times 4\) c) \(2 \times 10\) and \(4 \times 5\) d) \(3 \times 10\) and \(5 \times 6\)

- Find all factor pairs of \(30\). - Include both numbers from each factor pair. - Check that dividing \(30\) by each group size leaves no remainder.

1. Find every factor pair of \(30\): \(1 \times 30\), \(2 \times 15\), \(3 \times 10\), and \(5 \times 6\). 2. The possible group sizes are all the factors in those pairs: \(1, 2, 3, 5, 6, 10, 15\), and \(30\).

The possible group sizes are \(1, 2, 3, 5, 6, 10, 15\), and \(30\) students.

- List every factor of \(20\). - List every factor of \(30\). - Find the numbers that appear in both lists. - Factor pairs can help you make sure no factors are missing.

1. The factors of \(20\) are \(1, 2, 4, 5, 10\), and \(20\). 2. The factors of \(30\) are \(1, 2, 3, 5, 6, 10, 15\), and \(30\). 3. The numbers in both lists are \(1, 2, 5\), and \(10\).

The common factors of \(20\) and \(30\) are \(1, 2, 5\), and \(10\).

- Write \(72\) as products of two whole numbers. - Test possible factors in order. - Each factor pair gives you two factors. - You may stop once the factor pairs begin repeating in reverse order.

1. Find factor pairs systematically: \(1\times72\), \(2\times36\), \(3\times24\), \(4\times18\), \(6\times12\), and \(8\times9\). 2. Combine the numbers from the pairs and list them in increasing order: \(\{1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72\}\).

\(\{1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72\}\)

- Use division to test whether one number is a factor of another. - A factor divides a number with no remainder. - Is there a number that is a factor of every whole number? - Read statements containing “not” carefully.

1. Since \(72\div6=12\) with no remainder, statement a is true. 2. Since \(44\div8\) is not a whole number, statement b is false. 3. Since \(39\div13=3\), the number \(13\) is a factor of \(39\). Therefore, statement c is false. 4. The number \(1\) is a factor of every whole number, so statement d is true. 5. Since \(75\div25=3\), statement e is true.

a) True b) False c) False d) True e) True

- Find multiplication pairs whose product is the target number. - Include \(1\) and the number itself. - List the factor pairs systematically so none are missed. - Check whether each possible divisor leaves a remainder.

1. The factor pairs of \(15\) are \(1 \times 15\) and \(3 \times 5\), so its factors are \(1, 3, 5\), and \(15\). 2. The factor pairs of \(24\) are \(1 \times 24\), \(2 \times 12\), \(3 \times 8\), and \(4 \times 6\), so its factors are \(1, 2, 3, 4, 6, 8, 12\), and \(24\). 3. The factor pairs of \(28\) are \(1 \times 28\), \(2 \times 14\), and \(4 \times 7\), so its factors are \(1, 2, 4, 7, 14\), and \(28\). 4. The factor pairs of \(30\) are \(1 \times 30\), \(2 \times 15\), \(3 \times 10\), and \(5 \times 6\), so its factors are \(1, 2, 3, 5, 6, 10, 15\), and \(30\).

\(15\): \(1, 3, 5, 15\) \(24\): \(1, 2, 3, 4, 6, 8, 12, 24\) \(28\): \(1, 2, 4, 7, 14, 28\) \(30\): \(1, 2, 3, 5, 6, 10, 15, 30\)

- How can you organize factor pairs so that none are missed? - What does it mean for one number to be a factor of another? - Which division calculation can test the claim?

1. Find factor pairs systematically: \(1\times42\), \(2\times21\), \(3\times14\), and \(6\times7\). 2. The positive factors of \(42\) are \(\{1,2,3,6,7,14,21,42\}\). 3. Since \(42\div14=3\) with no remainder, \(14\) is a factor of \(42\).

The positive factors are \(\{1,2,3,6,7,14,21,42\}\). Yes, \(14\) is a factor of \(42\) because \(42\div14=3\).

- Think of multiplication facts and extend them using place value. - Try dividing each target number by small whole numbers. - Record both factors whenever the division has no remainder. - Make sure each equation uses a different factor pair.

1. For \(60\), possible factor pairs include \(2 \times 30\), \(3 \times 20\), \(4 \times 15\), \(5 \times 12\), and \(6 \times 10\). 2. For \(70\), the three factor pairs that do not use \(1\) are \(2 \times 35\), \(5 \times 14\), and \(7 \times 10\). 3. For \(100\), possible factor pairs include \(2 \times 50\), \(4 \times 25\), \(5 \times 20\), and \(10 \times 10\).

a) One possible answer is \(2 \times 30 = 60\), \(3 \times 20 = 60\), and \(5 \times 12 = 60\). b) \(2 \times 35 = 70\), \(5 \times 14 = 70\), and \(7 \times 10 = 70\) c) One possible answer is \(2 \times 50 = 100\), \(4 \times 25 = 100\), and \(10 \times 10 = 100\).

- Think of multiplication facts whose products can be scaled to \(96\). - Try halving \(96\). - If you double one factor and halve the other, the product stays the same.

1. Test small factors of \(96\). 2. Since \(96 \div 2 = 48\), one pair is \(2\) and \(48\). 3. Since \(96 \div 3 = 32\), another pair is \(3\) and \(32\). 4. Since \(96 \div 4 = 24\), another pair is \(4\) and \(24\). 5. Other valid pairs include \(6\) and \(16\).

One possible answer is \(2 \times 48\), \(3 \times 32\), and \(4 \times 24\).

- Count the rows and columns in each array. - Compare the difference between the two factors in each pair.

1. Array a) has \(4\) rows of \(6\) dots, so it matches \(4 \times 6\). 2. Array b) has \(3\) rows of \(8\) dots, so it matches \(3 \times 8\). 3. Array c) has \(2\) rows of \(12\) dots, so it matches \(2 \times 12\). 4. Array a) is the least stretched out because its factor pair, \(4\) and \(6\), is closest together.

a) \(4 \times 6\) b) \(3 \times 8\) c) \(2 \times 12\) Array a) is the most compact.

- Determine the total shown in each array. - Test each proposed factor against both totals. - Keep a number only if it works for both arrays.

1. Array a) has \(24\) dots. Every listed number divides \(24\): \(2,3,4,6,8,\) and \(12\). 2. Array b) has \(36\) dots. The listed numbers that divide \(36\) are \(2,3,4,6,\) and \(12\); \(8\) does not divide \(36\) evenly. 3. Therefore, the common factors are \(2,3,4,6,\) and \(12\).

The common factors are \(2,3,4,6,\) and \(12\).

- Use the number of columns as the divisor. - The quotient is the number of complete rows.

1. Array a) has \(12\) columns and \(4\) rows, so \(48 \div 12 = 4\). 2. Array b) has \(8\) columns and \(6\) rows, so \(48 \div 8 = 6\). 3. Array c) has \(6\) columns and \(8\) rows, so \(48 \div 6 = 8\). 4. The quotients are \(4, 6\), and \(8\).

a) \(48 \div 12 = 4\) b) \(48 \div 8 = 6\) c) \(48 \div 6 = 8\)

- A factor must have a whole-number partner whose product is \(48\). - A multiple can be written as a whole-number product of the given number. - Treat each statement independently.

1. True, because \(6\times8=48\). 2. False, because no whole-number multiple of \(7\) equals \(48\). 3. True, because \(96=2\times48\). 4. True, because \(48=4\times12\).

1) True, because \(6\times8=48\). 2) False, because \(48\) is not a whole-number multiple of \(7\). 3) True, because \(96=2\times48\). 4) True, because \(48=4\times12\).

- For each proposed row size, look for a whole-number partner factor. - A working row size must use every block with none left over. - Write a multiplication equation for each size that works.

1. A row size of \(6\) works because \(6\times12=72\). 2. A row size of \(8\) works because \(8\times9=72\). 3. A row size of \(9\) works because \(9\times8=72\). 4. A row size of \(10\) does not work because no whole number multiplied by \(10\) equals \(72\).

The possible row sizes are \(6\), because \(6\times12=72\); \(8\), because \(8\times9=72\); and \(9\), because \(9\times8=72\). A row size of \(10\) does not work.

- A factor pair uses two whole numbers whose product is \(12\). - Start with the smallest possible factor and work upward without skipping possibilities. - Decide whether changing only the order changes which two factors are used.

1. A factor pair is a pair of whole numbers whose product is the target number. 2. \(1\times12=12\), \(2\times6=12\), and \(3\times4=12\), so the factor pairs are \(1\) and \(12\), \(2\) and \(6\), and \(3\) and \(4\). 3. Reversing a multiplication, such as \(12\times1\), uses the same two factors and therefore does not create a new factor pair.

\(1\) and \(12\); \(2\) and \(6\); \(3\) and \(4\). Reversing the order does not make a new factor pair.

- Count rows and columns in each array. - Compare the two numbers you multiply in each picture. - Does swapping the order change the two factors themselves?

1. Array a) has \(3\) rows of \(6\), so it shows \(3 \times 6 = 18\). 2. Array b) has \(6\) rows of \(3\), so it shows \(6 \times 3 = 18\). 3. Both use the factors \(3\) and \(6\). Reversing row and column counts does not make a new factor pair.

The same factor pair, \((3,6)\), in reversed order.

- Start by listing factor pairs for one number. - Find the factor pairs of the other number independently. - Compare the complete lists rather than guessing from just one multiplication fact.

1. Factor pairs of \(18\) are \((1,18)\), \((2,9)\), and \((3,6)\), so its factors are \(1,2,3,6,9,18\). 2. Factor pairs of \(24\) are \((1,24)\), \((2,12)\), \((3,8)\), and \((4,6)\), giving \(1,2,3,4,6,8,12,24\). 3. The factors in both lists are \(1,2,3,6\).

The factor pairs of \(18\) are \((1,18)\), \((2,9)\), and \((3,6)\). For \(24\) they are \((1,24)\), \((2,12)\), \((3,8)\), and \((4,6)\). No new pair can have both factors \(5\) or greater, since \(5 \times 5 = 25\) exceeds both numbers. The shared factors are \(1,2,3,6\).

- Draw a rectangle for the proposed new pair. - How would it compare with a rectangle you already drew?

1. \((9,4)\) uses exactly the same two factors as \((4,9)\), so it is not new. 2. After \((6,6)\), a larger first factor would pair with a smaller second factor already listed. 3. Thus all different positive factor pairs of \(36\) are present.

No. \((9,4)\) reverses \((4,9)\). The list is complete: after \((6,6)\), a larger first factor would pair with a smaller factor already listed.

- How many different factor pairs does 40 have? - Use each factor pair as the length and width of a rectangle. - Find the perimeter of each rectangle. - Compare every perimeter with \(30\,\text{ft}\).

1. List all whole-number factor pairs of 40, representing the numbers of tiles along the two sides: \((1, 40), (2, 20), (4, 10), (5, 8)\). 2. Because each tile is \(1\,\text{ft}\) square, these factor pairs are also the rectangle dimensions in feet. 3. Compute each perimeter using \(P=2\times(l+w)\): - For \(1\times40\), \(P=2\times(1+40)=82\,\text{ft}\). - For \(2\times20\), \(P=2\times(2+20)=44\,\text{ft}\). - For \(4\times10\), \(P=2\times(4+10)=28\,\text{ft}\). - For \(5\times8\), \(P=2\times(5+8)=26\,\text{ft}\). 4. None of the possible perimeters is \(30\,\text{ft}\), so the requested arrangement is not possible.

No. The possible perimeters are \(82\,\text{ft}\), \(44\,\text{ft}\), \(28\,\text{ft}\), and \(26\,\text{ft}\), so none is \(30\,\text{ft}\).

- Generate factor pairs systematically from the smallest possible first factor upward. - Each factor pair contributes two factors to the complete list. - Stop when continuing would only reverse a pair you already found.

1. The positive factor pairs of \(48\) are \(1 \times 48\), \(2 \times 24\), \(3 \times 16\), \(4 \times 12\), and \(6 \times 8\). 2. Therefore, the positive factors are \(1,2,3,4,6,8,12,16,24,48\). 3. The factors greater than \(4\) but less than \(10\) are \(6\) and \(8\), so those are the possible row sizes in the requested interval. 4. The pairs are complete because after \(6 \times 8\), reversing the factors would only repeat pairs already listed.

Factor pairs: \(1\times48\), \(2\times24\), \(3\times16\), \(4\times12\), \(6\times8\). Positive factors: \(1,2,3,4,6,8,12,16,24,48\). Possible row sizes greater than \(4\) and less than \(10\): \(6\) and \(8\). The list of factor pairs is complete because after \(6\times8\), continuing with larger first factors would only reverse pairs already listed.

- Work through the numbers in order so you do not skip any. - For each number, find factor pairs and count all distinct positive factors. - Compare the counts in your table once every row is complete.

1. List or pair the positive factors of each number, counting a repeated middle factor only once. For example, \(12\) has factors \(1,2,3,4,6,12\), and \(16\) has factors \(1,2,4,8,16\). 2. The factor counts, in order from \(10\) through \(20\), are \(4,2,6,2,4,4,5,2,6,2,6\). 3. The largest count is \(6\), achieved by \(12\), \(18\), and \(20\).

a) <table><tr><th>Number</th><th>Positive factors</th></tr><tr><td>\(10\)</td><td>\(4\)</td></tr><tr><td>\(11\)</td><td>\(2\)</td></tr><tr><td>\(12\)</td><td>\(6\)</td></tr><tr><td>\(13\)</td><td>\(2\)</td></tr><tr><td>\(14\)</td><td>\(4\)</td></tr><tr><td>\(15\)</td><td>\(4\)</td></tr><tr><td>\(16\)</td><td>\(5\)</td></tr><tr><td>\(17\)</td><td>\(2\)</td></tr><tr><td>\(18\)</td><td>\(6\)</td></tr><tr><td>\(19\)</td><td>\(2\)</td></tr><tr><td>\(20\)</td><td>\(6\)</td></tr></table> b) \(12\), \(18\), and \(20\) each have \(6\) positive factors, the greatest count.

- Try small whole-number factors and find their partners. - Check each pair with multiplication. - If you switch two factors, does that make a new pair?

1. For \(48\), three possible pairs are \(2\times24\), \(3\times16\), and \(4\times12\). 2. For \(72\), three possible pairs are \(2\times36\), \(3\times24\), and \(8\times9\). 3. For \(84\), three possible pairs are \(2\times42\), \(3\times28\), and \(7\times12\). 4. Each factor pair multiplies to its target, and each chosen pair has a different set of factors.

a) \(2\times24\), \(3\times16\), \(4\times12\) b) \(2\times36\), \(3\times24\), \(8\times9\) c) \(2\times42\), \(3\times28\), \(7\times12\) Other correct factor pairs are possible.

- List and count all positive factors of \(48\). - Repeat the process for \(60\). - Compare the two counts. - Use factor pairs so that you do not miss any factors.

1. The factor pairs of \(48\) are \((1, 48)\), \((2, 24)\), \((3, 16)\), \((4, 12)\), and \((6, 8)\). Thus its positive factors are \(\{1, 2, 3, 4, 6, 8, 12, 16, 24, 48\}\), for a total of \(10\). 2. The factor pairs of \(60\) are \((1, 60)\), \((2, 30)\), \((3, 20)\), \((4, 15)\), \((5, 12)\), and \((6, 10)\). Thus its positive factors are \(\{1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60\}\), for a total of \(12\). 3. Since \(12>10\), the number \(60\) has more positive factors.

\(60\) has more positive factors. The positive factors of \(48\) are \(\{1, 2, 3, 4, 6, 8, 12, 16, 24, 48\}\), for a total of \(10\). The positive factors of \(60\) are \(\{1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60\}\), for a total of \(12\).

- First list all positive factors of \(60\). - Which numbers in your list are in the multiples-of-\(5\) pattern? - What ones digits do multiples of \(5\) have? - Check each factor one at a time.

1. The positive factors of \(60\) are \(1,2,3,4,5,6,10,12,15,20,30,\) and \(60\). 2. Select the factors that are multiples of \(5\). 3. The numbers are \(5,10,15,20,30,\) and \(60\).

\(5,10,15,20,30,60\)

- Find the factor pairs for each number separately. - Count the factors after each list is complete. - Remember that \(6 \times 6 = 36\), so \(6\) appears only once in the factor list. - Work upward systematically to avoid missing a factor pair.

1. The factors of \(21\) are \(1, 3, 7\), and \(21\), for a total of \(4\) factors. 2. The factors of \(36\) are \(1, 2, 3, 4, 6, 9, 12, 18\), and \(36\), for a total of \(9\) factors. 3. The factors of \(40\) are \(1, 2, 4, 5, 8, 10, 20\), and \(40\), for a total of \(8\) factors. 4. Since \(9 > 8 > 4\), the number \(36\) has the greatest number of factors.

a) Factors of \(21\): \(1, 3, 7, 21\) Factors of \(36\): \(1, 2, 3, 4, 6, 9, 12, 18, 36\) Factors of \(40\): \(1, 2, 4, 5, 8, 10, 20, 40\) b) \(36\) has the greatest number of factors.

- Check each whole number strictly between \(10\) and \(20\). - Prime numbers have exactly two positive factors. - Use factor pairs to make a complete list for each number.

1. Check each whole number from \(11\) through \(19\). 2. The numbers \(11, 13, 17,\) and \(19\) are prime, so each has only two positive factors. 3. The number \(12\) has \(\{1,2,3,4,6,12\}\), so it has six factors. 4. The number \(14\) has \(\{1,2,7,14\}\), so it has four factors. 5. The number \(15\) has \(\{1,3,5,15\}\), so it has four factors. 6. The number \(16\) has \(\{1,2,4,8,16\}\), so it has five factors. 7. The number \(18\) has \(\{1,2,3,6,9,18\}\), so it has six factors. 8. Therefore, the numbers are \(14\) and \(15\).

The numbers are \(14\), with positive factors \(\{1,2,7,14\}\), and \(15\), with positive factors \(\{1,3,5,15\}\).

- First find all positive factors of \(72\). - Use multiplication facts to find factor pairs such as \(8\times9\). - Then keep only the factors in the required interval.

1. Find the factor pairs of \(72\): \(1\times72\), \(2\times36\), \(3\times24\), \(4\times18\), \(6\times12\), and \(8\times9\). 2. The complete list of positive factors is \(1,2,3,4,6,8,9,12,18,24,36,72\). 3. Apply the condition \(7<d<15\). The factors that satisfy it are \(8,9,\) and \(12\).

\(8,9,\) and \(12\)

- What does it mean for a number to be a factor? - List the positive factors of each number separately. - Which values occur in both lists? - Check possible factors systematically, beginning with \(1\).

1. The positive factors of \(60\) are \(1,2,3,4,5,6,10,12,15,20,30,\) and \(60\). 2. The positive factors of \(84\) are \(1,2,3,4,6,7,12,14,21,28,42,\) and \(84\). 3. The numbers that appear in both lists are \(1,2,3,4,6,\) and \(12\).

\(1,2,3,4,6,12\)

- First consider which positive numbers can be factors of \(70\). - Check which of those factors also occur in the multiples-of-\(14\) pattern. - What is the greatest value you need to test?

1. The positive multiples of \(14\) up to \(70\) are \(14,28,42,56,\) and \(70\). 2. The positive factors of \(70\) are \(1,2,5,7,10,14,35,\) and \(70\). 3. The numbers that appear in both lists are \(14\) and \(70\). Multiples of \(14\) greater than \(70\) cannot be factors of \(70\).

\(14\) and \(70\)

- Find an equal factor pair of \(64\). - The dividend equals the divisor times the quotient. - Determine the combined effect of doubling both factors.

1. The divisor and quotient form an equal factor pair of \(64\). Since \(8 \times 8 = 64\), the equation is \(64 \div 8 = 8\). 2. Doubling both values gives \(16\) and \(16\). The new dividend is \(16 \times 16 = 256\). 3. Since \(256 \div 64 = 4\), the dividend is multiplied by \(4\).

a) \(64 \div 8 = 8\) b) The new dividend is \(256\), which is \(4\) times the original dividend.

- Use factor pairs to make a complete list for each number. - Put each list in order, count its entries, and compare the counts.

1. The positive factors of \(24\) are \(\{1,2,3,4,6,8,12,24\}\), so \(24\) has \(8\) factors. 2. The positive factors of \(30\) are \(\{1,2,3,5,6,10,15,30\}\), so \(30\) has \(8\) factors. 3. The positive factors of \(36\) are \(\{1,2,3,4,6,9,12,18,36\}\), so \(36\) has \(9\) factors. 4. Since \(9>8\), the number \(36\) has the greatest number of positive factors.

\(24\) has \(8\) positive factors: \(\{1,2,3,4,6,8,12,24\}\). \(30\) has \(8\) positive factors: \(\{1,2,3,5,6,10,15,30\}\). \(36\) has \(9\) positive factors: \(\{1,2,3,4,6,9,12,18,36\}\). Therefore, \(36\) has the greatest number of positive factors.

- First list the positive multiples of \(12\) that are less than \(50\). - Find all positive factors of each multiple systematically. - Compare the factor counts in an organized list or table.

1. The positive multiples of \(12\) less than \(50\) are \(12,24,36,\) and \(48\). 2. Their positive factor sets and counts are: - \(12: \{1,2,3,4,6,12\}\), with \(6\) factors. - \(24: \{1,2,3,4,6,8,12,24\}\), with \(8\) factors. - \(36: \{1,2,3,4,6,9,12,18,36\}\), with \(9\) factors. - \(48: \{1,2,3,4,6,8,12,16,24,48\}\), with \(10\) factors. 3. Therefore, \(48\) has the greatest number of positive factors.

\(12\) has \(6\) factors, \(24\) has \(8\), \(36\) has \(9\), and \(48\) has \(10\). Therefore, \(48\) has the greatest number of positive factors.

- Count rows and dots per row in each rectangle. - A complete rectangular array shows a multiplication fact. - The row and column counts are factors of the total.

1. Array a) represents \(2\times12=24\), array b) represents \(3\times8=24\), and array c) represents \(4\times6=24\). 2. Each rectangle arranges the same \(24\) dots into different numbers of equal rows and columns, so the products are equal. 3. The visible side lengths give the factors \(2,3,4,6,8,12\).

a) \(2\times12=24\); \(3\times8=24\); \(4\times6=24\) b) Each array has the same number of dots arranged in different rows and columns. c) \(2,3,4,6,8,12\)

- Find the planned total using the rows and columns. - Count the unfilled spots and subtract to find those present. - Look for two whole-number factors of the remaining number.

1. The planned array has \(7\) rows of \(8\) dots, for \(7\times8=56\) students. 2. Seven dots are unfilled, leaving \(56-7=49\) students. 3. Since \(49=7\times7\), the remaining students can form a complete \(7\)-by-\(7\) array.

Yes. The \(49\) students who are present can stand in \(7\) rows of \(7\).

- The number of groups must be a factor of \(60\). - Apply the “no more than \(12\)” condition after finding possible group counts. - Use the chosen group count to determine the group size.

1. The factors of \(60\) that are no greater than \(12\) are \(1,2,3,4,5,6,10,\) and \(12\). 2. The greatest allowed factor is \(12\), so the maximum is \(12\) groups. 3. Since \(12\times5=60\), each group has \(5\) dots.

The maximum is \(12\) groups with \(5\) dots in each group.

- Start with whole-number factor pairs of the total. - A factor pair can be used in either row-column direction. - Check both the row limit and the muffins-per-row limit.

1. Factor pairs of \(24\) are \((1,24)\), \((2,12)\), \((3,8)\), and \((4,6)\). 2. Check both arrangements for each pair, keeping more than \(2\) and at most \(8\) rows, and fewer than \(6\) muffins per row. 3. The possibilities are \(6\) rows of \(4\) or \(8\) rows of \(3\).

\(6\) rows with \(4\) muffins each, or \(8\) rows with \(3\) muffins each.

- Write the two factor lists in order and compare them. - List multiples of both numbers until the first match appears. - For part c, compare the greatest possible factor with the least positive multiple.

1. The positive factors of \(12\) are \(1,2,3,4,6,12\). The positive factors of \(18\) are \(1,2,3,6,9,18\). Their common factors are \(1,2,3,\) and \(6\). 2. The positive multiples of \(10\) begin \(10,20,30,40,\ldots\), and the positive multiples of \(15\) begin \(15,30,45,\ldots\). The least common positive multiple is \(30\). 3. Every positive factor of \(20\) is at most \(20\), and every positive multiple of \(20\) is at least \(20\). Therefore, the only number in both groups is \(20\).

a) Factors of \(12\): \(1,2,3,4,6,12\) Factors of \(18\): \(1,2,3,6,9,18\) Common factors: \(1,2,3,6\) b) \(30\) c) Yes, \(20\).

- List the whole numbers strictly between \(20\) and \(30\). - Find their positive factors systematically by using factor pairs. - Which number has exactly three distinct positive factors?

1. Check the numbers from \(21\) through \(29\). 2. The prime numbers \(23\) and \(29\) each have only two positive factors. The composite candidates \(21\), \(22\), \(24\), \(26\), \(27\), and \(28\) each have at least four positive factors: \(21\) has \(1, 3, 7, 21\); \(22\) has \(1, 2, 11, 22\); \(24\) has more than three; \(26\) has \(1, 2, 13, 26\); \(27\) has \(1, 3, 9, 27\); and \(28\) has more than three. 3. The number \(25\) has the positive factors \(1, 5,\) and \(25\), exactly three factors. 4. Therefore, the number is \(25\), and its complete set of positive factors is \(\{1, 5, 25\}\).

The number is \(25\). Its positive factors are \(\{1, 5, 25\}\).

- Use factor pairs to find all positive factors without missing any. - Try writing factors in pairs whose product is the number. - Prime numbers cannot work because they have only two positive factors. - Consider what happens to the factor count for perfect squares.

1. Check numbers in the interval by listing factor pairs. 2. One valid choice is \(12\), whose positive factors are \(\{1,2,3,4,6,12\}\). It has exactly six factors. 3. A second valid choice is \(18\), whose positive factors are \(\{1,2,3,6,9,18\}\). It also has exactly six factors. 4. Other valid choices are \(20\), with \(\{1,2,4,5,10,20\}\), and \(28\), with \(\{1,2,4,7,14,28\}\).

One possible answer is \(12\) and \(18\). The positive factors of \(12\) are \(\{1,2,3,4,6,12\}\). The positive factors of \(18\) are \(\{1,2,3,6,9,18\}\). The numbers \(20\) and \(28\) are also valid choices.

- Organize each factor list in pairs whose product is the number. - Compare a pair such as \(4\times4\) with a pair such as \(2\times8\). - Ask what happens to the count when the two members of one factor pair are the same.

1. The positive factors are: \(4:\{1,2,4\}\), \(9:\{1,3,9\}\), \(16:\{1,2,4,8,16\}\), and \(25:\{1,5,25\}\). Their factor counts are \(3,3,5,3\). 2. The positive factors are: \(6:\{1,2,3,6\}\), \(8:\{1,2,4,8\}\), and \(10:\{1,2,5,10\}\). Each has \(4\) factors. 3. Factors can be paired so that each pair has product equal to the number. For a perfect square, one pair uses the same factor twice, such as \(4\times4=16\), so that middle factor is counted once. All other factors occur in distinct pairs. A positive non-square has only distinct factor pairs. Therefore perfect squares have odd factor counts and positive non-squares have even factor counts.

a) \(4\) has \(3\) factors, \(9\) has \(3\), \(16\) has \(5\), and \(25\) has \(3\). b) Each of \(6,8,\) and \(10\) has \(4\) factors. The perfect squares have odd factor counts, while these non-squares have even factor counts. c) Factor pairs are distinct except for the equal-factor pair in a perfect square, so that one factor is counted only once. Thus positive perfect squares have odd factor counts and positive non-squares have even factor counts.