Find the remainder when \(3 \times 13 \times 23 \times 33 \times \ldots \times 183 \times 193\) is divided by 5.
(3×13×23×33×...×183×193) = product_(n=1)^20(-7 + 10 n) = 9936127455089061347552058319626135201 mod 5 =1
(0 + 3) (10 + 3) =
(30 + 9)(20 + 3)=
(600 + 180 + 90 + 27) (30 + 3) =
(..............+ 81)(40 + 3)....... etc.
So (3 x 13 x 23 x........183 x 193) / 5 will have a remainder =
3^20 mod 5 =
(3^4)^5 mod 5 =
(3^4) mod 5 =
1
+4620 
