Suppose 1 ≤ x, y, z ≤ 9 are integers such that x > z. Let α and β be the three-digit numbers given by α = xyz and β = zyx so that x, y, z are the digits as they are shown. Set γ = α − β. How many of the following statements are always true?
2 does not divide γ.
3 divides γ.
5 does not divide γ.
7 does not divide γ.
11 divides γ.
13 does not divide γ.
Select the correct answer.
- A4
- B3
- C5
- D6
Explanation
γ = (100x + 10y + z) − (100z + 10y + x) = 99(x − z) = 9 × 11 × (x − z), with 1 ≤ x − z ≤ 8. Since 99 is divisible by 3 and by 11, '3 divides γ' and '11 divides γ' are always true. Since x − z ≤ 8, 13 never divides γ, so '13 does not divide γ' is always true. But γ can be even (x − z even), a multiple of 5 (x − z = 5) or of 7 (x − z = 7), so the other three claims fail. Exactly 3 statements are always true.