Perfect Squares' Squares' Properties

Number Theory Level 3

$n$ is any integer. Denote the sum of all the possible unit digits of $n^4$ as $S$ . Find $504\cdot S^{-1}$ .

Details & Assumptions :

$S^{-1}$ denotes the multiplicative inverse.

The answer is 42.

1 solution

Alex Delhumeau
Jun 3, 2015

If $n$ is an integer, then its possible unit digit values are $0, 1,2,3,4,5,6,7,8,$ and $9$ .

Hence, $n^2$ 's possible unit digits are $0, 1, 4, 5, 6,$ and $9$ .

Therefore, $n^4$ has possible unit digits of $0, 1, 5,$ and $6$ .

$504 * \frac{1}{1+5+6}=\boxed{42}$ , the Ultimate Answer to the Ultimate Question!

Ultimate ? Why so ?

A Former Brilliant Member - 6 years ago

It's a reference to The Hitchhiker's Guide to the Galaxy by Douglas Adams, where the answer to the "Ultimate Question of Life, the Universe, and Everything" is $42$ .

Alex Delhumeau - 6 years ago

Oh! Right , how could I forget that ! :(

A Former Brilliant Member - 6 years ago

Perfect Squares' Squares' Properties

The answer is 42.

1 solution

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