Questions: Patterns in Addition and Multiplication

Question 1

A student is checking whether 47 is a multiple of 9 using the digit-sum pattern. She adds 4 + 7 = 11. What should she conclude?

• A. 47 is a multiple of 9 because 11 is close to 9
• B. 47 is not a multiple of 9 because 11 does not equal 9 or a multiple of 9
• C. She needs to check if 47 ends in 7 before deciding
• D. The digit-sum pattern only works for numbers less than 40

Answer: (B)

The 9s digit-sum pattern says: the digits of any multiple of 9 sum to 9 (or a multiple of 9). 4 + 7 = 11, which is neither 9 nor a multiple of 9, so 47 is NOT a multiple of 9. (Verify: 9 × 5 = 45, 9 × 6 = 54 — 47 is between them.) 'Close to 9' is not the same as 'equals 9.' The pattern gives a definitive test, not an approximation.

Question 2

Why do all multiples of 5 end in either 0 or 5, without exception?

• A. It's a coincidence that happens to be true for small numbers but breaks down for large ones
• B. Multiples of 5 end in 5 or 0 because of a rule someone invented to make the pattern easier
• C. Multiplying by 5 is repeated addition of 5, and adding 5 makes the ones digit cycle: 5, 0, 5, 0...
• D. Only odd multiples of 5 end in 5; even multiples end in 0

Answer: (C)

Multiples of 5 are 5, 10, 15, 20, 25... — you're adding 5 each time. Starting from 5: ones digit is 5. Add 5 → ones digit is 0. Add 5 → ones digit is 5 again. The ones digit alternates between 5 and 0 forever, because adding 5 to a number ending in 5 always gives a number ending in 0, and vice versa. This is a consequence of how base-10 arithmetic works, not a coincidence.

Question 3

Every multiple of 5 ends in either 0 or 5 — no exceptions, no matter how large the number.

• T. True
• F. False

Answer: True

This pattern holds for all multiples of 5 without exception, because it follows from the structure of base-10 arithmetic. When you repeatedly add 5, the ones digit alternates perfectly between 5 and 0: 5, 10, 15, 20... 995, 1000, 1005... The pattern never breaks because the ones digit only depends on the ones digits of the numbers being added, and those cycle perfectly.

Question 4

Patterns in multiplication tables (like the 5s ending in 0 or 5) are just memory tricks — they don't reflect any real mathematical reason.

• T. True
• F. False

Answer: False

These patterns are mathematical consequences, not arbitrary tricks. They arise directly from two facts: (1) multiplication is repeated addition, and (2) our base-10 number system makes ones digits cycle when you add. The 5s pattern exists because adding 5 repeatedly makes ones digits cycle between 5 and 0 — that's a mathematical fact about how numbers work. Understanding why makes the pattern far more memorable and useful.

Question 5

Why do the digits of multiples of 9 always sum to 9 (or a multiple of 9), and how can you use this to check if a number is a multiple of 9?

Model answer: In base 10, each time you add 9 to a number, the tens digit increases by 1 and the ones digit decreases by 1 — keeping the digit sum constant. To check if a number is a multiple of 9, add its digits: if they sum to 9 (or a multiple of 9), the number is a multiple of 9; if not, it isn't.

For example: 9 (digit sum 9), 18 (1+8=9), 27 (2+7=9), 36 (3+6=9), 45 (4+5=9). The pattern holds for large numbers too: 99 (9+9=18, 1+8=9), 108 (1+0+8=9). This is a real mathematical property of 9 in base 10, not a memorization shortcut — though it doubles as one of the most useful mental math checks a student can learn.