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Question

Prove that for every positive integer n, 1n+8n3n6n is divisible by 10.

Solution
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Since 10 is the product of two primes 2 and 5, it will suffice to show that the given expression is divisible both by 2 and 5. To do so, we shall use the simple fact that if a and b be any positive integers, then anbn is always divisible by ab.
Writting A1n+8n3n6n
=(8n3n)(6n1n)
we find that 8n3n and 6n1n are both divisible by 5, and consequently A is by 5(=83=61). Again, writing (8n6n)(3n1n), we find that A is divisible by 2(=86=31). Hence A is divisible by 10.

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