538

Front 1

a_1 cong a_2 mod b iff ___

Back 1

b divides a_1 minus a_2

Front 2

a_1 mod b plus a_2 mod b cong ___

Back 2

a_1 plus a_2 mod b

Front 3

a_1 mod b plus a_2 mod b cong a_1 plus a_2 mod b because sum of remainders ___

Back 3

less sum of quantities is divisible by b

Front 4

a_1 mod b times a_2 mod b cong a_1 times a_2 because prod of rems ___

Back 4

less prod of quantities cancels out rems and b factors out, so result is divisible by b

Front 5

neg a cong b minus a because ___

Back 5

subtracting b minus a from minus a gives quant divis by b

Front 6

neg a cong b minus a rule allows ___

Back 6

computing with negatives for simplicity, if their abs values are smaller