777 base

Number Theory Level 1

$N$ is an integer whose representation in base $b$ is 777. Find the smallest positive integer $b$ for which $N$ is the fourth power of an integer.

The answer is 18.

2 solutions

A Former Brilliant Member
Jan 22, 2016

Let $N=k^{4}$ for some positive integer $k$ . Then, we get $7b^{2}+7b+7=k^{4}$ $7\mid k^{4} \Rightarrow 7\mid k$ Substituting in $k=7$ , we get the answer $b=18$ . (You can check the discriminant as well, but I left it out.)

How did you conclude its 7?, i didn't get it.

Shivam Singh - 4 years, 10 months ago

Good solution (I did it by brute force, translating 777 in various bases into base 10 until I had a number that I knew was a fourth power - I am good at rapid fire calculations, so it did not take l.ong)

Thomas Sutcliffe - 3 years, 4 months ago

We have $7b^2+7b+7=k^4\text{ or }7(b^2+b+1)=k^4\text{ then } 7|k^4 \rightarrow 7|k$

Tran Hieu - 5 years, 4 months ago

Sorry for the long and convoluted solution...

A Former Brilliant Member - 5 years, 4 months ago

Les Schumer
Jul 27, 2020

Let $N = k^4$ for some positive integer $k$ . Then, we get $\\ 7b^2 + 7b + 7 = k^4\\ 7(b^2 + b + 1) = k^4\\$ $k^4$ has $7$ as a factor, so is of a form $7^4*x^4 \\=> 7^4*x^4 = 7(b^2 + b + 1) \\=> 7^3\; |\; (b^2 + b + 1) \\ => (7^3-1)\;|\;b(b+1)\\ => 342\;|\;b(b+1)\\$ By observation, $342=18*19$ so we have $b=\boxed{18}$

777 base

The answer is 18.

2 solutions

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