Divisibility Patterns: Homework

Seven quick tests for which numbers divide evenly, without dividing.

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Part A — Digit Sums and Last Digits / 5

Find the number each rule asks for, then answer.

Test each number against 2, 3, 4, 5, 6, 9 and 10. List the divisors that work, with commas between.

Each problem has a mistake or a trap in it.

1.
A student says 5,126 is divisible by 4 because 6 is even. The last two digits are 26. Is 26 divisible by 4? Type yes or no.Answer
2.
A student says 4,015 is divisible by 3 because it ends in 5. What is the real digit sum, and is it divisible by 3?Digit sumDivisible by 3
3.
A student says 4,130 is divisible by 6 because it's even. Is it? Type yes or no.Answer
4.
Write a number that is divisible by 3 but not by 9, and show how you know.Answer

These numbers are real. Test each against 2, 3, 4, 5, 6, 9 and 10.

Write your answer in a sentence or two.

1.
Write a four-digit number that is divisible by 2, 3, 4, 5, 6, 9 and 10 all at once. How did you build it?Answer
2.
Is every multiple of 9 also a multiple of 3? Is every multiple of 3 also a multiple of 9? Explain.Answer

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Total / 21

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Notes

Part A — the answers.

Part B — the divisors that work.

2, 3, 4, 5, 6, 9, 10. The ones digit is 0, the digit sum is 9, and 20 ÷ 4 = 5.
2, 3, 4, 6, 9. The digit sum is 18 and 16 ÷ 4 = 4, but it doesn't end in 0 or 5.
5. It ends in 5, but it is odd and the digit sum is 14.
2, 3, 6, 9. The digit sum is 9, but 02 is not divisible by 4.
None. It's odd, the digit sum is 14, and it doesn't end in 0 or 5.

Part C — the answers.

No. 26 ÷ 4 = 6 remainder 2, so 5,126 is not divisible by 4.
10. No. The ones digit has nothing to do with the rule for 3.
No. The digit sum is 8, so it fails the rule for 3, and 6 needs both 2 and 3.
Answers vary. One that works is 2,001: the digit sum is 3, which divides by 3 and not by 9.

Part D — the divisors that work.

a. 2, 3, 4, 5, 6, 10. The digit sum is 15, so it fails for 9. b. 5 only. c. All of them: 2, 3, 4, 5, 6, 9, 10. d. 2, 4, 5, 10. The digit sum is 14, so it fails for 3, 6 and 9. e. 7

Part E — one correct answer, and what a good explanation says.

Answers vary. Make it end in 00 so it passes 2, 4, 5 and 10, then make the digits add to 9 so it passes 3, 6 and 9, such as 3,600 or 4,500.
Answers vary. Yes to the first, since 9 is 3 times 3. No to the second, since 6 is a multiple of 3 and not of 9.