A Problem of Power

Number Theory Level 4

What can we say is always true about the sum of two numbers $x^n$ and $x^m$ where $n,m$ and $x$ are all integers with $n,m\geq 0$ and $x>1$ ?

None of these choices A perfect square A power of

x

Never divisible by

x

A multiple of

x

An even number

1 solution

William Whitehouse
May 10, 2016

What yo need to consider is the case when either n or m is 0:

In this case we have x^m+1 which is odd when x is even, not a multiple of x we can see that it's not a perfect square or a power of x just by subbing in some values (e.g. x=2, m=2 x^m+1=5 which isn't square nor is this a power of x).

All that is left is to show that there are values of n and m such that the sum is divisible by x, again we can do this by substitution. x=2,m=1,n=2 ==> 2^2+2^2=6 and 6 is a multiple of 2.

The only remaining option is none of these

A Problem of Power

1 solution

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