Question

after the division of a number successively by 3 4 and 7 the remainders obtained are 2 1 and 4 respectively. what will be the remainder if 84 divides the same number?

Answer 1

Then,

x≡2mod3

x≡1mod4

x≡4mod7

Since 3,4,7 are pairwise co-prime. We can apply Chinese remainder theorem.

x=((2×(4×7)×((4×7)−1mod3))+(1×(3×7)×((3×7)−1mod4))+(4×(3×4)×((3×4)−1mod7)))mod(3×4×7)

⟹x=((2×28×(28−1mod3))+(1×21×(21−1mod4))+(4×12×(12−1mod7)))mod84

⟹x=((2×28×1)+(1×21×1)+(4×12×3))mod84

⟹x=(56+21+144)mod84

⟹x=221mod84

⟹x=53.