What's C and G?

Number Theory Level 5

Let C C be the answer to this problem and G G be the answer to this problem .

What is the last digit of 2 C + 3 C + … + G C 2^C + 3^C + \ldots + G^C ?


The answer is 4.

Ivan Koswara by Ivan Koswara , 25, Indonesia

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1 solution

Mark Hennings
Apr 11, 2016

In these three linked problems, we are looking for the solutions of the equations 2 C + 3 C + ⋯ + G C ≡ N ( m o d 10 ) ( G − N N ) = C G = 1 6 C N \begin{array}{rcl} 2^C + 3^C + \cdots + G^C & \equiv & N \pmod{10} \\ {G-N \choose N} & = & C \\ G & = & \tfrac16CN \end{array} Since N N must be a single digit integer, it is easy to check that the solution to these simultaneous equations is N = 4 N=\boxed{4} , C = 15 C=15 and G = 10 G=10 .

5 Helpful 0 Interesting 0 Brilliant 0 Confused

The idea of making a trio of problems was amazingly genius!

Also, nice solution!

Shourya Pandey - 5 years, 2 months ago

How is it easy to check? It's easy to do a bit of trial and error, but can you find a more elegant way of solving it?

Stewart Gordon - 5 years, 2 months ago

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