Chinese Remainder Theorem (CRT) Youtube Lecture Handout
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Recap
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4 Types of HCF Remainder Problems
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4 Types of LCM Remainder Problems
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Smallest Number When Divided by x, y and z Leaves Remainder a, b, c?
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x - a = y - b = z - c = common difference d
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Smallest number divided by 2, 3, 4, 5, 6 leaves remainder 1, 2, 3, 4, 5
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Find the smallest number which when divided by 2, 3 and 5 produces 1, 2, 3 as remainders
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Note that 2, 3 and 5 are (pairwise) relatively co-prime
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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!!
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Find the smallest number which when divided by 2, 3 and 5 produces 1, 2, 2 as remainders
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Find the smallest number which when divided by 2, 3 and 5 produces 1, 2, 3 as remainders
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Find the smallest number which when divided by 7, 9 and 11 produces 1, 2, 3 as remainders
One Small Concept
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Constant Case: Smallest number when divided by x, y and z leaves remainder a, b, c
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x - a = y - b = z - c = common difference d
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LCM (a, b, c) – d
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Convert to Constant Case: Smallest number when divided by x, y and z leaves remainder a, b, c, where
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x - ma = y - mb = z - mc = common difference d
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Use all combining
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24 produces a remainder 4 when divided by 5
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Find the smallest number which when divided by 7, 9 and 11 produces 1, 2, 3 as remainders
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7 – 1 = 5 7 – 2 × 1 = 5
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9 – 2 = 7 9 – 2 × 2 = 5
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11 – 3 = 8 11 – 2 × 3 = 5
CRT is Last Resort!!
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Constant Case: Smallest number when divided by x, y and z leaves remainder a, b, c
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x - a = y - b = z - c = common difference d
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LCM (a, b, c) – d
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Convert to Constant Case: Smallest number when divided by x, y and z leaves remainder a, b, c, where
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x - ma = y - mb = z - mc = common difference d
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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