Question

a number is divisible by 3 if the sum of its digits is divisible by 3. a number is divisible by 9 if the sum of its digits is divisible by 9. 4. circle the numbers that are divisible by 9. 46 55 72 197 213 3670 5004 11348 5. circle the numbers that are divisible by 3. 10 15 45 52 300 525 3721 8730 42125 6. a) underline the numbers in question 5 that are divisible by both 3 and 9. bonus > why is it not possible for a number to be divisible by 9 but not divisible by 3? 7. a) sort the whole numbers from 0 to 30 into the venn diagram below. venn - diagram with multiples of 2 and multiples of 3 circles. b) in which region of the venn diagram are the numbers that are divisible by 6? why did that happen? c) use the tests for divisibility by 2 (end in an even number) and 3 (sum of the digits is a multiple of 3) to make a test for divisibility by 6. d) circle the multiples of 6. 72 88 941 642 153 2340 4500 6903 8. find and list all the factors of the number using divisibility tests. a) 35 1, 5, 7, 35 b) 18 c) 28 d) 48 e) 54 f) 76 9. ava has 174 marbles that she wants to divide into smaller groups with none left over. ava wants to put the same number of marbles in each bag. circle the number of bags that ava can use. 2 bags 3 bags 5 bags 6 bags 9 bags

Explanation:

Step1: Recall divisibility rules

A number is divisible by 9 if the sum of its digits is divisible by 9. A number is divisible by 3 if the sum of its digits is divisible by 3. A number is divisible by 6 if it is divisible by both 2 (ends in an even - number) and 3.

Step2: Find numbers divisible by 9 in question 4

- For 46: \(4 + 6=10\), not divisible by 9.
- For 55: \(5 + 5 = 10\), not divisible by 9.
- For 72: \(7+2 = 9\), divisible by 9.
- For 197: \(1+9 + 7=17\), not divisible by 9.
- For 213: \(2 + 1+3 = 6\), not divisible by 9.
- For 3870: \(3+8 + 7+0=18\), divisible by 9.
- For 5004: \(5+0 + 0+4 = 9\), divisible by 9.
- For 11348: \(1+1+3 + 4+8=17\), not divisible by 9.

Step3: Find numbers divisible by 3 in question 5

- For 10: \(1+0 = 1\), not divisible by 3.
- For 15: \(1+5 = 6\), divisible by 3.
- For 45: \(4 + 5=9\), divisible by 3.
- For 52: \(5+2 = 7\), not divisible by 3.
- For 300: \(3+0+0 = 3\), divisible by 3.
- For 525: \(5+2+5 = 12\), divisible by 3.
- For 3721: \(3+7+2 + 1=13\), not divisible by 3.
- For 8730: \(8+7+3+0 = 18\), divisible by 3.
- For 42125: \(4+2+1+2 + 5=14\), not divisible by 3.

Step4: Find numbers divisible by both 3 and 9 in question 6

From question 5, the numbers divisible by both 3 and 9 are 45 and 8730 (since their digit - sums are divisible by 9 and thus also by 3).

Step5: Answer bonus question

If a number is divisible by 9, since \(9=3\times3\), the sum of its digits is divisible by 9. Since 9 is a multiple of 3, if the sum of digits is divisible by 9, it is also divisible by 3. So it's not possible for a number to be divisible by 9 but not by 3.

Step6: Sort numbers in Venn - diagram in question 7a

Multiples of 2 from 0 to 30: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30.
Multiples of 3 from 0 to 30: 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30.
Numbers in the intersection (multiples of both 2 and 3, i.e., multiples of 6): 0, 6, 12, 18, 24, 30.
Numbers only in multiples of 2: 2, 4, 8, 10, 14, 16, 20, 22, 26, 28.
Numbers only in multiples of 3: 3, 9, 15, 21, 27.

Step7: Answer question 7b

The numbers divisible by 6 are in the intersection of the "Multiples of 2" and "Multiples of 3" circles in the Venn - diagram because 6 is the least common multiple of 2 and 3, so a number must be a multiple of both 2 and 3 to be a multiple of 6.

Step8: Answer question 7c

A number is divisible by 6 if it ends in an even number (divisible by 2) and the sum of its digits is a multiple of 3 (divisible by 3).

Step9: Find multiples of 6 in question 7d

- For 72: Ends in 2 (even) and \(7 + 2=9\) (divisible by 3), so it is a multiple of 6.
- For 88: Ends in 8 (even) but \(8+8 = 16\) (not divisible by 3), so it is not a multiple of 6.
- For 941: Ends in 1 (not even), so it is not a multiple of 6.
- For 642: Ends in 2 (even) and \(6 + 4+2=12\) (divisible by 3), so it is a multiple of 6.
- For 153: Ends in 3 (not even), so it is not a multiple of 6.
- For 2340: Ends in 0 (even) and \(2+3 + 4+0=9\) (divisible by 3), so it is a multiple of 6.
- For 4500: Ends in 0 (even) and \(4+5+0+0 = 9\) (divisible by 3), so it is a multiple of 6.
- For 6903: Ends in 3 (not even), so it is not a multiple of 6.

Step10: Find factors in question 8

- For 18: \(18=1\times18=2\times9 = 3\times6\), factors are 1, 2, 3, 6, 9, 18.
- For 28: \(28=1\times28=2\times14 = 4\times7\), factors are 1, 2, 4, 7, 14, 28.
- For 48: \(48=1\times48=2\times24=3\times16 = 4\times12=6\times8\), factors are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
- For 54: \(54=1\times54=2\times27=3\times18 = 6\times9\), factors are 1, 2, 3, 6, 9, 18, 27, 54.
- For 76: \(76=1\times76=2\times38 = 4\times19\), factors are 1, 2, 4, 19, 38, 76.

Step11: Answer question 9

For 174:
- Check divisibility by 2: 174 ends in 4 (even), so it is divisible by 2.
- Check divisibility by 3: \(1+7 + 4=12\), divisible by 3.
- Check divisibility by 5: 174 does not end in 0 or 5, so it is not divisible by 5.
- Check divisibility by 6: Since it is divisible by 2 and 3, it is divisible by 6.
- Check divisibility by 9: \(1+7+4 = 12\), not divisible by 9.
The number of bags Ava can use are 2 bags, 3 bags, 6 bags.

Answer:

1. Numbers divisible by 9: 72, 3870, 5004.
2. Numbers divisible by 3: 15, 45, 300, 525, 8730.
3. Numbers divisible by both 3 and 9: 45, 8730.

7a. Refer to the above - mentioned sorting of numbers in Venn - diagram.
7b. In the intersection of "Multiples of 2" and "Multiples of 3" because 6 is the LCM of 2 and 3.
7c. A number is divisible by 6 if it ends in an even number and the sum of its digits is a multiple of 3.
7d. Multiples of 6: 72, 642, 2340, 4500.

1. 18: 1, 2, 3, 6, 9, 18; 28: 1, 2, 4, 7, 14, 28; 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48; 54: 1, 2, 3, 6, 9, 18, 27, 54; 76: 1, 2, 4, 19, 38, 76.
2. 2 bags, 3 bags, 6 bags.