Question

If n is positive integer, show that 3^{n+2}-2^{n+2}+3^n-2^n is divisible by 10

Collected in the board: Number Theory

Steven Zheng posted 2 days ago

Answer

3^{n+2}-2^{n+2}+3^n-2^n

If n=1

3^{1+2}-2^{1+2}+3^1-2^1

=27-8+3-2 = 20

which is divisible by 10

If n>1

3^{n+2}-2^{n+2}+3^n-2^n

=3^{n+2}+3^n-(2^{n+2}+2^n)

=3^n(3^2+1)-2^n(2^2+1)

=10\cdot 3^n-5\cdot 2^n

=10\cdot 3^n-10\cdot 2^{n-1}

which is divisible by 10

Therefore

3^{n+2}-2^{n+2}+3^n-2^n is divisible by 10 for any positive integer n

Steven Zheng posted 2 days ago

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