MathLabs

Problem 2

Find one pair of positive integers a,ba,b such that ab(a+b)ab(a+b) is not divisible by 77, but (a+b)7−a7−b7(a+b)^7-a^7-b^7 is divisible by 777^7.
Step 2 of 3: Choose an explicit pair
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In plain words

The small choice b=1b=1 turns the quadratic into a2+a+1a^2+a+1, and a=18a=18 makes it the exact required power of seven.

a=18,b=1,ab(a+b)=342,a2+ab+b2=343=73a=18,\quad b=1,\quad ab(a+b)=342,\quad a^2+ab+b^2=343=7^3
Detailed analysis

For this pair, ab(a+b)=18⋅1⋅19=342ab(a+b)=18\cdot1\cdot19=342 is not divisible by 77, while a2+ab+b2=324+18+1=343=73a^2+ab+b^2=324+18+1=343=7^3.