Free math worksheets

- 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 \(60\).

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

- 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 \(99\div9=11\).

- 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 \(4\) 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 \(48\).

\(12, 24, 36, 48\)

- 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 the multiples of \(8\) in the interval. - List the multiples of \(12\) in the interval. - Select the numbers that appear in both lists.

1. The multiples of \(8\) in the interval are \(24, 32, 40, 48, 56, 64, 72, 80, 88\), and \(96\). 2. The multiples of \(12\) in the interval are \(24, 36, 48, 60, 72, 84\), and \(96\). 3. The common multiples are \(24, 48, 72\), and \(96\). 4. For example, \(3 \times 8 = 24\) and \(2 \times 12 = 24\); the same check works for each listed number.

\(24, 48, 72, 96\)

- 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 \(56\).

\(56\)

- Choose a number of chairs and subtract their legs from \(38\). - Check whether the remaining number is a multiple of \(3\). - Try different numbers of chairs.

1. With \(2\) chairs, the chairs have \(2 \times 4 = 8\) legs. The remaining \(38 - 8 = 30\) legs make \(30 \div 3 = 10\) stools. 2. With \(5\) chairs, the chairs have \(5 \times 4 = 20\) legs. The remaining \(38 - 20 = 18\) legs make \(18 \div 3 = 6\) stools.

Two possible combinations are: - \(10\) stools and \(2\) chairs - \(6\) stools and \(5\) chairs

- Try the possible numbers of tricycles systematically. - Subtract the tricycle wheels from \(18\). - Check whether the remaining wheels can be divided into groups of \(2\).

1. Try positive numbers of tricycles and check whether the remaining wheels can form bicycles. 2. With \(2\) tricycles, \(2 \times 3 = 6\) wheels are used. The remaining \(12\) wheels make \(12 \div 2 = 6\) bicycles. 3. With \(4\) tricycles, \(4 \times 3 = 12\) wheels are used. The remaining \(6\) wheels make \(6 \div 2 = 3\) bicycles. 4. Six tricycles use all \(18\) wheels and leave no bicycles, so that case is excluded. These are all the possible combinations.

The two possible combinations are: - \(6\) bicycles and \(2\) tricycles - \(3\) bicycles and \(4\) tricycles

- 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 \(54\)

- What does it mean for a number to be a multiple of \(18\)? - Make a list of consecutive multiples of \(18\). - Check both boundary conditions carefully. - Which is the first multiple greater than the lower bound?

1. Generate multiples of \(18\): \(18, 36, 54, 72, 90, 108, \ldots\). 2. Apply the condition \(40<x<100\). 3. The multiples that satisfy both inequalities are \(54, 72,\) and \(90\).

\(54, 72,\) and \(90\)

- 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 \(96\).

\(24,48,72,96\)

- 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 \(96\)

- How can you generate consecutive multiples of \(15\)? - Check both strict boundaries carefully. - How is a multiple different from a factor?

1. Generate multiples of \(15\): \(15,30,45,60,75,90,105,\ldots\). 2. Apply the condition \(40<x<100\). The numbers are \(45,60,75,\) and \(90\). 3. The greatest number in the set is \(90\).

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

- 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.

- Write the product as one fraction. - Determine what factor must be supplied to cancel the denominator \(8\). - List the multiples of that factor in the allowed range. - Check the neighboring multiples just outside the range to confirm completeness.

1. The product is \(\frac{3n}{8}\). 2. Since \(3\) has no factor of \(2\), \(n\) must supply all three factors of \(2\) in the denominator \(8\). Therefore, \(n\) must be a multiple of \(8\). 3. The multiples of \(8\) from \(1\) through \(24\) are \(8,16,24\). 4. The next multiples outside the range are \(0\) and \(32\), so the list contains every allowed value. The valid values increase by \(8\).

\(n=8,16,24\); the valid values are multiples of \(8\)

- Organize your attempts in a table. - Choose a number of large towers and find the blocks remaining. - Check whether the remainder can be divided into groups of \(4\).

1. Try even numbers of large towers so the remaining blocks can be grouped by \(4\). 2. With \(2\) large towers, \(2 \times 6 = 12\) blocks are used. The remaining \(48 - 12 = 36\) blocks make \(36 \div 4 = 9\) small towers. 3. With \(4\) large towers, \(4 \times 6 = 24\) blocks are used. The remaining \(24\) blocks make \(24 \div 4 = 6\) small towers. 4. With \(6\) large towers, \(6 \times 6 = 36\) blocks are used. The remaining \(12\) blocks make \(12 \div 4 = 3\) small towers.

Three combinations are: - \(9\) small towers and \(2\) large towers - \(6\) small towers and \(4\) large towers - \(3\) small towers and \(6\) large towers