smc

Question

Find the units digit of the decimal expansion of (15 + 220)^19 + (15 + 220)^82.

Accepted Answer

Let A=15+220 and B=15-220. Note that A^19+B^19 and A^82+B^82 are both integers: When we expand (Binomial Theorem) and combine like terms for each expression, the rational terms are added and the irrational terms are canceled. We have align* A^19+B^19 &= [19015^19220^0+19115^18220^1++191915^0220^19] + [19015^19220^0-19115^18220^1+-191915^0220^19] &= 2[19015^19220^0+19215^17220^2++191815^1220^18] &= 2[19015^19+19215^17220++191815^1 220^9]. align* Similarly, we have A^82+B^82=2[82015^82+82215^80220++8282220^41]. We add the two equations and take the sum modulo 10: align* (A^19+A^82)+(B^19+B^82) &= 2[19015^19+ 19215^17220++191815^1 220^9_010 ]+2[82015^82+ 82215^80220++8282220^41_010 ] & 2[19015^19]+2[82015^82] & 2[5]+2[5] & 010. align* It is clear that 0<B^82<B^19<B<0.5, from which 0<B^19+B^8…