- Recall the multiples of \(6\). - Check whether each number can be divided by \(6\) with no remainder. - Test every number in the list.

1. Check the multiples of \(6\) in the list: \(6 \times 2 = 12\), \(6 \times 3 = 18\), \(6 \times 5 = 30\), \(6 \times 7 = 42\), \(6 \times 8 = 48\), and \(6 \times 10 = 60\). 2. Therefore, the listed multiples of \(6\) are \(12, 18, 30, 42, 48\), and \(60\).

\(12, 18, 30, 42, 48, 60\)

- Start at the tick for zero and move right one tick at a time. - Add the same amount at each tick. - Count how many equal jumps you needed to reach P.

1. Moving from \(0\) to P takes four equal jumps. 2. Count \(4, 8, 12, 16\). Therefore P is \(16\), the fourth multiple of \(4\).

Four jumps; P is \(16\).

- Add the same jump size each time, beginning at zero. - Keep track of the landing numbers rather than estimating the distance. - Stop when the next jump would pass the target.

1. Adding \(6\) each time gives \(6,12,18,24,30\). 2. The next landing is \(36\), past \(32\), so no whole number of these jumps lands on \(32\).

\(6,12,18,24,30\). No; the next jump lands on \(36\), not \(32\).

- One pack adds the same number of markers each time. - What totals can you make with whole packs? - Look for a multiplication fact with the pack size.

1. Each unopened pack contributes \(6\) markers. 2. Since \(6 \times 4 = 24\), four packs give exactly \(24\). Thus \(24\) is a multiple of \(6\).

Yes. Buy \(4\) packs because \(6 \times 4 = 24\).

- Begin with one group of seven. - Add another equal group to find each new multiple. - Check that each term is a whole-number product of seven.

1. Start at \(7\) and add \(7\) each time. 2. The first four positive multiples are \(7,14,21,28\).

\(7,14,21,28\); add \(7\) each time.

- List several positive multiples of \(5\) and \(10\). - Look at the ones digit of each multiple. - Compare the two lists to find the values they share.

1. The positive multiples of \(5\) are \(5, 10, 15, 20, 25, 30, \ldots\). Their ones digits alternate between \(5\) and \(0\). 2. Every positive multiple of \(10\) has a ones digit of \(0\). 3. Every multiple of \(10\) is also a multiple of \(5\). Through \(100\), the common multiples are \(10, 20, 30, 40, 50, 60, 70, 80, 90\), and \(100\).

a) \(0\) and \(5\) b) \(0\) c) \(10, 20, 30, 40, 50, 60, 70, 80, 90, 100\)

- List multiples of \(8\) in order. - Keep only the multiples inside the stated interval. - A number with remainder \(5\) is \(5\) more than a multiple of \(8\). - Stop when the next number would be greater than \(40\).

1. For a), continue the multiples of \(8\): \(8 \times 6 = 48\), \(8 \times 7 = 56\), \(8 \times 8 = 64\), \(8 \times 9 = 72\), \(8 \times 10 = 80\), and \(8 \times 11 = 88\). 2. For b), add \(5\) to consecutive multiples of \(8\): \(0 + 5 = 5\), \(8 + 5 = 13\), \(16 + 5 = 21\), \(24 + 5 = 29\), and \(32 + 5 = 37\).

a) \(48, 56, 64, 72, 80, 88\) b) \(5, 13, 21, 29, 37\)

- How are the multiples of a number related to its multiplication facts? - Which inverse operation can test whether a number is a multiple of \(9\)?

1. Multiply \(9\) by the whole numbers \(1\) through \(5\): \(9\times1=9\), \(9\times2=18\), \(9\times3=27\), \(9\times4=36\), and \(9\times5=45\). 2. Test \(99\) by division: \(99\div9=11\). Because the quotient is a whole number, \(99\) is a multiple of \(9\).

The first five positive multiples are \(9, 18, 27, 36,\) and \(45\). Yes, \(99\) is a multiple of \(9\) because \(99\div9=11\).

- Generate multiples by adding \(18\) repeatedly. - Keep track of both the lower and upper limits. - Values equal to either boundary would not qualify.

1. Count by \(18\): \(18,36,54,72,90,108,\dots\). 2. Keep only those greater than \(40\) and less than \(100\): \(54,72,90\).

\(54,72,90\)

- For each statement, ask whether the first number can be written as the second number times a whole number. - A single multiplication equation can justify a true statement. - Pay attention to statements that include the word “not.”

1. Statement a) is true because \(9\times16=144\). 2. Statement b) is false because \(9\times6=54\), so \(54\) is a multiple of \(9\). 3. Statement c) is true because \(7\times9=63\). 4. Statement d) is false because \(19\) cannot be written as \(2\) times a whole number. 5. Statement e) is false because \(5\times19=95\).

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

- List the multiples of both numbers. - Include only values less than \(100\). - Mark the values that appear in both lists. - Look for a pattern among the common multiples.

1. The positive multiples of \(8\) less than \(100\) are \(8,16,24,32,40,48,56,64,72,80,88,\) and \(96\). 2. The positive multiples of \(12\) less than \(100\) are \(12,24,36,48,60,72,84,\) and \(96\). 3. The numbers in both lists are \(24,48,72,\) and \(96\).

\(24,48,72,96\)

- List the multiples of \(4\) in the interval. - List the multiples of \(6\) in the interval. - Find the numbers that appear in both lists.

1. The multiples of \(4\) in the interval are \(12, 16, 20, 24, 28, 32, 36, 40, 44\), and \(48\). 2. The multiples of \(6\) in the interval are \(12, 18, 24, 30, 36, 42\), and \(48\). 3. The numbers in both lists are \(12, 24, 36\), and \(48\).

\(12, 24, 36, 48\)

- Generate a series of multiples of \(15\). - Keep only numbers strictly between the bounds. - Find the greatest value after you have the complete set.

1. Count by \(15\): \(15,30,45,60,75,90,105,\dots\). 2. Within the required range, the multiples are \(45,60,75,90\). 3. The greatest of these is \(90\).

a) \(\{45,60,75,90\}\) b) \(90\)

- Think about the products you get by multiplying \(8\) by whole numbers. - Check each candidate against the sequence of multiples of \(8\). - Make sure every selected number can be written as \(8\) times a whole number.

1. A multiple of \(8\) can be written as \(8\times n\) for a whole number \(n\). 2. From the list, \(16=8\times2\), \(24=8\times3\), \(32=8\times4\), \(48=8\times6\), \(56=8\times7\), \(64=8\times8\), \(72=8\times9\), and \(80=8\times10\). 3. Therefore, those eight numbers are the multiples of \(8\) in the list.

\(16,24,32,48,56,64,72,80\)

- List multiples of \(7\) near the stated interval. - List multiples of \(8\) near the stated interval. - Find the number that appears in both lists.

1. The multiples of \(7\) near the interval are \(49, 56\), and \(63\). 2. The multiples of \(8\) near the interval are \(48, 56\), and \(64\). 3. The only number in both lists that is greater than \(50\) and less than \(70\) is \(56\).

\(56\)

- Look for ticks that have a marker in both panels. - Ignore zero when you choose the first positive match. - Check the matching number using both skip-counting patterns.

1. Line a) lands at \(4,8,12,16,20,24\). 2. Line b) lands at \(6,12,18,24\). 3. The first shared positive landing point is \(12\).

\(12\). Both lines mark \(12\), since \(4 \times 3 = 12\) and \(6 \times 2 = 12\). There is no smaller positive mark shared by both.

- Use the labeled numbers to work out each marked point. - Count by sixes and see which positions match. - Check whether any point falls between two consecutive multiples.

1. The line marks A at \(18\), B at \(22\), and C at \(30\). 2. Both \(18\) and \(30\) are multiples of \(6\): \(6 \times 3 = 18\) and \(6 \times 5 = 30\). 3. No whole-number multiple of \(6\) equals \(22\), so point B is not a multiple.

Point B at \(22\). The other points are \(18 = 6 \times 3\) and \(30 = 6 \times 5\), but \(22\) lies between \(18\) and \(24\), two consecutive multiples of \(6\).

- Decide which number is the size of each equal group. - A multiple of twenty must be made from whole groups of twenty. - Try multiplying twenty by a small positive whole number.

1. Since \(4 \times 5 = 20\), \(4\) is a factor of \(20\), not a multiple of \(20\). 2. A multiple of \(20\) is \(20\) times a whole number, such as \(20 \times 2 = 40\).

\(4\) is a factor of \(20\), not a multiple. One multiple is \(40\).

- Write the early flash times for each lamp. - Find the times that appear on both lists. - The question asks for two shared times after the start, not just the first one.

1. Common positive multiples of \(4\) and \(10\) begin at \(20\) seconds. 2. The next is \(40\) seconds, since each lamp has completed a whole number of its cycles.

\(20\) seconds and \(40\) seconds.

- Only whole multiples of the per-round score are possible. - Find a multiple just below the target. - Move to the next multiple and check whether it crosses the target.

1. Multiples of \(9\) near \(50\) are \(9\times5=45\) and \(9\times6=54\). 2. The first one exceeding \(50\) is \(54\).

\(54\) points, from \(9 \times 6\).

- Write the multiples of \(15\) in order. - Use \(15 \times 10\) as a benchmark if it helps you track the pattern. - A number with remainder \(4\) is \(4\) more than a multiple of \(15\).

1. For a), the multiples of \(15\) in the interval are \(15 \times 1 = 15\), \(15 \times 2 = 30\), \(15 \times 3 = 45\), \(15 \times 4 = 60\), \(15 \times 5 = 75\), and \(15 \times 6 = 90\). 2. For b), add \(4\) to multiples of \(15\): \(30 + 4 = 34\), \(45 + 4 = 49\), \(60 + 4 = 64\), and \(75 + 4 = 79\).

a) \(15, 30, 45, 60, 75, 90\) b) \(34, 49, 64, 79\)

- Write the two multiplication patterns one below the other. - Look for the first number that occurs in both lists.

1. The first eight positive multiples of \(5\) are \(5,10,15,20,25,30,35,\) and \(40\). 2. The first eight positive multiples of \(8\) are \(8,16,24,32,40,48,56,\) and \(64\). 3. The first number that appears in both lists is \(40\).

a) Multiples of \(5\): \(5,10,15,20,25,30,35,40\) Multiples of \(8\): \(8,16,24,32,40,48,56,64\) b) \(40\)

- What totals do you get by repeatedly adding \(6\)? - In part b, remember that each total must satisfy both conditions: greater than \(40\) and less than \(60\).

1. The possible totals are the positive multiples of \(6\): \(6, 12, 18, 24, \ldots\). 2. Check the multiples near the given interval: \(6\times6=36\), which is too small; \(6\times7=42\); \(6\times8=48\); \(6\times9=54\); and \(6\times10=60\), which is not less than \(60\). 3. Therefore, the totals strictly between \(40\) and \(60\) are \(42, 48,\) and \(54\).

a) \(6, 12, 18, 24, \ldots\) b) \(42, 48,\) and \(54\)

- First list the multiples of \(8\) in the stated interval. - Test each listed number for divisibility by \(3\). - A digit sum can help you check divisibility by \(3\).

1. The multiples of \(8\) greater than \(50\) and less than \(100\) are \(56, 64, 72, 80, 88\), and \(96\). 2. Test those numbers for divisibility by \(3\). The numbers \(72\) and \(96\) are divisible by \(3\); the other listed multiples of \(8\) are not. 3. Therefore, the possible secret numbers are \(72\) and \(96\).

\(72\) and \(96\)

- Subtract the amount remaining from the starting amount. - Compare the amount used with nearby multiples of \(3\). - Equal whole-number shares among \(3\) bottles are possible only when the amount used is a multiple of \(3\).

1. The bottles received \(12\,\text{L}-5\,\text{L}=7\,\text{L}\) altogether. 2. Seven is not a multiple of \(3\): the nearby multiples are \(6=3\times2\) and \(9=3\times3\). 3. Therefore, \(7\,\text{L}\) cannot be split into \(3\) equal whole-number amounts.

\(7\,\text{L}\) were used. Since \(7\) is not a multiple of \(3\), the bottles could not all receive the same whole number of liters.