Let F_k(a, b)=(a+b)^k-a^k-b^k and let S= 1,2,3,4,5,6,7,8,9,10 . For how many ordered pairs (a, b) with a, b S and a b is F₅(a, b) F₃(a, b) an integer?

Step-by-step solution

aligned F ₅( a , ~b ) F ₃( a , ~b ) & = ( a + b )^5- a ^5- b ^5 ( a + b )^3- a ^3- b ^3 & = 5 3 ( a ^2+ b ^2+ ab ) and a & =3 k ₁ or 3 k ₁+1 or 3 k ₁+2 ~b & =3 k ₂ or 3 k ₂+1 or 3 k ₂+2 aligned only when a and b give same remainder a ^2+ b ^2+ ab is divisible by 3 . array lll & array ll a=1, & b=1,4,7,10 a=2, & b=2,5,8 array & 4 a=3, & b=3,6,9 & 3 a=4, & b=4,7,10 & 3 a=5, & b=5,8 & 2 a=6, & b=6,9 & 2 a=7, & b=7,10 & 2 a=8, & b=8 & 1 a=9, & b=9 & 1 a=10, & b=10 & 1 & Total & 22 ordered pair array