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9-ending numbers (Posted on 2019-04-29)

A and B are two positive numbers that end in 9. Prove that A²-B² is divisible by 40.

No Solution Yet Submitted by Danish Ahmed Khan

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| Comment 1 of 2

Let A=10a-1 and B=10b-1

then A^2-B^2=100a^2-20a-100b^2+20b

now 100a^2-20a is clearly divisible by 20 but upon factoring we get 20(5a^2-a)

But 5a^2-a is always even, so 100a^2-20a is divisible by 40 as is 100b^2-20b.

And so is their difference.

Posted by Jer on 2019-04-29 13:32:56

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