Without evaluation of it, prove that the number
N = 27,195^8 - 10,887^8 + 10,152^8 is divisible by
26,460.
Note: the original problem mistakenly listed the last number as 26,640. This has been corrected
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Submitted by pcbouhid
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Rating: 3.6667 (3 votes)
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Solution:
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The proof will be given in two steps.
(1) N = 27,195^8 - (10,887^8 - 10,152^8)
27,195 = (3 x 5 x 7^2 x 37), and so this number is divisible by (5 x 7^2).
The difference in the parentheses is divisible by (10,887 - 10,152) = 735 = (3 x 5 x 7^2), since (a^2n - b^2n) is divisible by (a - b) and so, also divisible by (5 x 7^2).
So, N is divisible by (5 x 7^2).
(2) N = (27,195^8 - 10,887^8) + 10,152^8
10,152 = (2^3 x 3^3 x 47), so, is divisible by (2^3 x 3^3).
The difference in parentheses is divisible by (27,195 - 10,887) = 16,308 = (2^2 x 3^3 x 151), so is divisible by (2^2 x 3^3).
So, N is divisible by (2^2 x 3^3).
Since N is divisible by (5 x 7^2) and by (2^2 x 3^3), it follows that N is divisible by the product of these numbers, because they are relative primes.
So, N is divisible by (2^2 x 3^3 x 5 x 7^2), which is 26,460.
One curious fact (it is ?) is that 27,295 - 10,887 + 10,152 = 26,460.
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