Chinese Remainder Theorem (CRT) Youtube Lecture Handout
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Recap

4 Types of HCF Remainder Problems

4 Types of LCM Remainder Problems

Smallest Number When Divided by x, y and z Leaves Remainder a, b, c?

x  a = y  b = z  c = common difference d

Smallest number divided by 2, 3, 4, 5, 6 leaves remainder 1, 2, 3, 4, 5

Find the smallest number which when divided by 2, 3 and 5 produces 1, 2, 3 as remainders

Note that 2, 3 and 5 are (pairwise) relatively coprime

Find the smallest number which when divided by 7, 9 and 11 produces 1, 2, 3 as remainders
Simplifying CRT
Key to reducing complicated calculations Not using Full CRT at all!!

Find the smallest number which when divided by 2, 3 and 5 produces 1, 2, 2 as remainders

Find the smallest number which when divided by 2, 3 and 5 produces 1, 2, 3 as remainders

Find the smallest number which when divided by 7, 9 and 11 produces 1, 2, 3 as remainders
One Small Concept

Constant Case: Smallest number when divided by x, y and z leaves remainder a, b, c

x  a = y  b = z  c = common difference d

LCM (a, b, c) – d


Convert to Constant Case: Smallest number when divided by x, y and z leaves remainder a, b, c, where

x  ma = y  mb = z  mc = common difference d



Use all combining

24 produces a remainder 4 when divided by 5

Find the smallest number which when divided by 7, 9 and 11 produces 1, 2, 3 as remainders

7 – 1 = 5 7 – 2 × 1 = 5

9 – 2 = 7 9 – 2 × 2 = 5

11 – 3 = 8 11 – 2 × 3 = 5
CRT is Last Resort!!

Constant Case: Smallest number when divided by x, y and z leaves remainder a, b, c

x  a = y  b = z  c = common difference d

LCM (a, b, c) – d


Convert to Constant Case: Smallest number when divided by x, y and z leaves remainder a, b, c, where

x  ma = y  mb = z  mc = common difference d



Use all combining
Next Class Word Problems on Chinese Remainder Theorem
6 professors begin courses of lectures on Monday, Tuesday, Wednesday, Thursday, Friday and Saturday, and announce their intentions of lecturing at intervals of 2, 3, 4, 1, 6, 5 days respectively. The regulations of the University forbid Sunday lectures. When first will all six professors simultaneously find themselves compelled to omit a lecture?
Mayank