If an 8-digit number 9 4 7 1 x 9 y 2 is divisible by 72, then which of the…
2024
If an 8-digit number 9 4 7 1 x 9 y 2 is divisible by 72, then which of the following statements is not true?
- A.
x = 8 and y = 5
- B.
x = 4 and y = 9
- C.
x = 9 and y = 5
- D.
x = 3 and y = 1
Attempted by 2 students.
Show answer & explanation
Correct answer: C
A number is divisible by 72 exactly when it is divisible by both 8 and 9, since 72 = 8 x 9 and these two factors share no common factor. Divisibility by 8 depends only on the number formed by the last three digits; divisibility by 9 depends on whether the sum of all the digits is a multiple of 9.
The 8-digit number is 9 4 7 1 x 9 y 2, so its last three digits are 9, y, 2 -- the value 900 + 10y + 2.
For divisibility by 8, 900 + 10y + 2 must be a multiple of 8. Testing y = 0 through 9: only y = 1 (912 divided by 8 = 114), y = 5 (952 divided by 8 = 119), and y = 9 (992 divided by 8 = 124) give an exact division; every other value of y leaves a remainder.
For divisibility by 9, the digit sum 9 + 4 + 7 + 1 + x + 9 + y + 2 = 32 + x + y must be a multiple of 9.
Each option gives one (x, y) pair; check every pair against both conditions to see which one fails -- that pair is the statement that is not true.
Option (x, y) | Last three digits (9y2) | Division by 8 | Digit sum (32+x+y) | Division by 9 | Both exact? |
|---|---|---|---|---|---|
x = 8, y = 5 | 952 | 952 divided by 8 = 119 | 45 | 45 divided by 9 = 5 | Yes |
x = 4, y = 9 | 992 | 992 divided by 8 = 124 | 45 | 45 divided by 9 = 5 | Yes |
x = 9, y = 5 | 952 | 952 divided by 8 = 119 | 46 | 46 divided by 9 = 5 remainder 1 | No |
x = 3, y = 1 | 912 | 912 divided by 8 = 114 | 36 | 36 divided by 9 = 4 | Yes |
Only the pair x = 9, y = 5 fails the digit-sum test for divisibility by 9, since 46 divided by 9 leaves remainder 1. So this is the statement that is not true, matching the saved answer.