37^22+210^22-44 when divided by 18 leaves the remainder Answer: 15

Question

$3	imes7^{22}+2	imes10^{22}-44$ when divided by 18 leaves the remainder

Answer: 15

Accepted Answer

$$3\times7^{22}+2\times10^{22}-44 = 3(6+1)^{22}+2(9+1)^{22}-44$$

$(1+6)^{22}=1+6\lambda$ and $(1+9)^{22}=1+9\mu$ for integers $\lambda,\mu$ (all higher terms carry factors of 6, 9 respectively).

$$=3(1+6\lambda)+2(1+9\mu)-44=18\lambda+3+18\mu+2-44=18\delta-39=18\delta'+15$$

Remainder = 15

Solutions · 1