Free Math Worksheets

- For factors, look for pairs of whole numbers whose product is the given number. - Divisibility rules for \(2\), \(3\), and \(5\) can help you test possible factors. - To generate multiples, multiply the number by \(1, 2, 3, \ldots\).

1. Pair whole numbers whose product is \(56\): \(1\times56\), \(2\times28\), \(4\times14\), and \(7\times8\). Therefore, the positive factors are \(\{1, 2, 4, 7, 8, 14, 28, 56\}\). 2. Pair whole numbers whose product is \(81\): \(1\times81\), \(3\times27\), and \(9\times9\). Therefore, the positive factors are \(\{1, 3, 9, 27, 81\}\). 3. Multiply \(17\) by \(1, 2, 3, 4,\) and \(5\). The first five positive multiples are \(17, 34, 51, 68,\) and \(85\). 4. Multiply \(18\) by \(1, 2, 3, 4,\) and \(5\). The first five positive multiples are \(18, 36, 54, 72,\) and \(90\).

a) \(\{1, 2, 4, 7, 8, 14, 28, 56\}\) b) \(\{1, 3, 9, 27, 81\}\) c) \(\{17, 34, 51, 68, 85\}\) d) \(\{18, 36, 54, 72, 90\}\)

- 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.

- Test each proposed factor with both totals. - Include a number only when both divisions have no remainder.

1. Every listed number divides \(24\): \(2, 3, 4, 6, 8\), and \(12\). 2. 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 divide the number with no remainder. - A multiple can be written as a whole-number product of the given number.

1. True, because \(48 \div 6 = 8\). 2. False, because \(48 \div 7\) has a remainder. 3. True, because \(96 = 2 \times 48\). 4. True, because \(48 = 4 \times 12\).

1) True, because \(48 \div 6 = 8\). 2) False, because \(48 \div 7\) has a remainder. 3) True, because \(96 = 2 \times 48\). 4) True, because \(48 = 4 \times 12\).

- For each proposed row size, look for a whole-number partner factor. - Use the ones digit to test the row size of \(10\).

1. A row size of \(6\) works because \(6 \times 12 = 72\). 2. A row size of \(8\) works because \(8 \times 9 = 72\). 3. A row size of \(9\) works because \(9 \times 8 = 72\). 4. A row size of \(10\) does not work because \(72 \div 10 = 7\) remainder \(2\).

The possible row sizes are \(6\), because \(6 \times 12 = 72\); \(8\), because \(8 \times 9 = 72\); and \(9\), because \(9 \times 8 = 72\). A row size of \(10\) leaves a remainder of \(2\).

- 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 \(30\,\text{ft}\).

- What is always the greatest positive factor of a number? - Recall how many positive factors a prime number has. - Test whether \(96\) can be written as \(x\) times a whole number.

1. The greatest positive factor of a number is the number itself, so \(x=48\). 2. A prime number has exactly two positive factors. The prime numbers in the set are \(2\) and \(3\); \(1\) is not prime, and every other number in the set has more than two positive factors. 3. Check whether \(96\) is a whole-number multiple of \(48\): \(48\times2=96\). Therefore, \(96\) is a multiple of \(x\).

a) \(x=48\) b) \(2\) and \(3\) c) Yes. Since \(48\times2=96\), the number \(96\) is a multiple of \(x\).

- For each number, test smaller whole numbers systematically to see which divide it with no remainder. - Use factor pairs. For example, if \(12\div2=6\), then both \(2\) and \(6\) are factors of \(12\). - Remember that every prime number has exactly two positive factors.

1. List the positive factors of each number and count them: \(10: \{1, 2, 5, 10\}\), so it has \(4\) factors. \(11: \{1, 11\}\), so it has \(2\) factors. \(12: \{1, 2, 3, 4, 6, 12\}\), so it has \(6\) factors. \(13: \{1, 13\}\), so it has \(2\) factors. \(14: \{1, 2, 7, 14\}\), so it has \(4\) factors. \(15: \{1, 3, 5, 15\}\), so it has \(4\) factors. \(16: \{1, 2, 4, 8, 16\}\), so it has \(5\) factors. \(17: \{1, 17\}\), so it has \(2\) factors. \(18: \{1, 2, 3, 6, 9, 18\}\), so it has \(6\) factors. \(19: \{1, 19\}\), so it has \(2\) factors. \(20: \{1, 2, 4, 5, 10, 20\}\), so it has \(6\) factors. 2. The greatest count is \(6\). The numbers \(12\), \(18\), and \(20\) each have \(6\) positive factors.

a) For \(10\) through \(20\), the numbers of positive factors are \(4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6\), respectively. b) The numbers \(12\), \(18\), and \(20\) have the greatest number of positive factors, with \(6\) each.

- 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 the rows and columns in each array. - The two side lengths of a complete rectangular array form a factor pair. - Reversing two factors does not create a new factor pair.

1. Array a) represents \(2 \times 12 = 24\), array b) represents \(3 \times 8 = 24\), and array c) represents \(4 \times 6 = 24\). 2. Each rectangle arranges the same \(24\) dots into a different number of equal rows and columns, so the arrays show different factor pairs of \(24\). 3. The visible side lengths give the factors \(2, 3, 4, 6, 8\), and \(12\).

a) \(2 \times 12 = 24\); \(3 \times 8 = 24\); \(4 \times 6 = 24\) b) The arrays show different factor pairs of the same number, \(24\). c) \(2, 3, 4, 6, 8, 12\)

- Find the number of students who remain. - Look for a factor pair of that number.

1. Find the number of students originally expected: \(7 \times 8 = 56\). 2. Subtract the absent students: \(56 - 7 = 49\). 3. Since \(49 = 7 \times 7\), the remaining students can form a square array with \(7\) rows and \(7\) columns.

Yes. The \(49\) remaining students can stand in a \(7 \times 7\) array.

- Find the greatest factor of \(60\) that is no greater than \(12\). - Then divide \(60\) by that number of groups.

1. Find the factors of \(60\) that are no greater than \(12\): \(1, 2, 3, 4, 5, 6, 10\), and \(12\). 2. The greatest allowed factor is \(12\), so the maximum is \(12\) groups. 3. Each group contains \(60 \div 12 = 5\) dots.

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

- 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.

- Write the factors in pairs whose product is the number. - What is different about the factor pair for \(16\) that contains \(4\)? - Think about whether a non-square can have a factor paired with itself.

1. List and count the positive factors: \(4: \{1, 2, 4\}\), so it has \(3\) factors. \(9: \{1, 3, 9\}\), so it has \(3\) factors. \(16: \{1, 2, 4, 8, 16\}\), so it has \(5\) factors. \(25: \{1, 5, 25\}\), so it has \(3\) factors. \(6: \{1, 2, 3, 6\}\), so it has \(4\) factors. \(8: \{1, 2, 4, 8\}\), so it has \(4\) factors. \(10: \{1, 2, 5, 10\}\), so it has \(4\) factors. 2. In these examples, each perfect square has an odd number of positive factors, while each non-square has an even number. 3. Positive factors usually form distinct pairs \((a, b)\) with \(a\times b=n\). For a perfect square, the pair containing its square root has equal entries, so the square root is counted only once. That unpaired factor makes the total odd. A non-square has only pairs of distinct factors, so its total is even.

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) Factors pair to make the number. A perfect square has one pair in which both factors are its square root, so that factor is counted once; all other factors occur in distinct pairs. Thus perfect squares have odd factor counts, and non-squares have even factor counts.