A Larger Divisibility Difference

Algebra Level 2

If we take a certain 3-digit integer and reverse its digits to form another 3-digit integer, the absolute difference between these two numbers is always divisible by which of the following numbers?

97 98 99 96

Chung Kevin by Chung Kevin , 45, Australia

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1 solution

Let the three digit number be A B C ‾ = 100 A + 10 B + C \overline{ABC}=100A+10B+C .Then the number formed by reversing its digits will be C B A ‾ = 100 C + 10 B + A \overline{CBA}=100C+10B+A .Their absolute difference would be: ∣ A B C ‾ − C B A ‾ ∣ = ∣ ( 100 A + 10 B + C ) − ( 100 C + 10 B + A ) ∣ = ∣ 99 A − 99 C ∣ = 99 ∣ A − C ∣ \begin{aligned} | \overline{ABC}-\overline{CBA} |&=|(100A+10B+C)-(100C+10B+A)|\\ &=|99A-99C|=99|A-C|\end{aligned} Therefore we can see that the difference of a three digit number and the number formed by reversing its digits is always divisible by 99 \boxed{99}

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