Cracking LCM & HCF Remainder Problems: 8 Simple Formulas Explained YouTube Lecture Handouts
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Watch Video Lecture on YouTube: Cracking LCM & HCF Remainder Problems: 8 Simple Formulas Explained
Cracking LCM & HCF Remainder Problems: 8 Simple Formulas Explained
Remainder Problems with HCF and LCM
Recap
Method:

Prime Factorization

Divisibility tests
Problem Keywords

LCM: Minimum number, least amount, smallest duration etc.

HCF: Maximum number, most amount, longest duration etc.
HCF Remainder Problems
4 Types
Greatest Number Which Divides X, Y and Z? HCF Type 1 (Simple)
Let’s start with a simple example: 12, 18, and 30
Greatest Number Which Divides X, Y and Z Leaves Same Remainder R (Given)?HCF Type 2 (Same Remainder Given)
Let’s find a number which divides all 14, 20 and 32 leaving remainder 2
Greatest Number Which Divides X, Y and Z Leaves Same Remainder R (Not Given)?HCF Type 3 (Same Remainder Not Given)
Number which divides all 14, 20 and 32 leaving same remainder
Greatest Number Which Divides X, Y and Z, Leaving Remainders a, B and C (Respectively) HCF Type 4 (Different Remainder Given)
Number which divides 12, 18 and 30 leaving remainder 2, 3 and 0.
HCF Problems – 4 Types Summary Understand and Remember

Greatest number which divides x, y and z = HCF (x, y, z)

Greatest number which divides x, y and z and leaves remainder r = HCF(x  r, y  r, z  r)

Greatest number which divides x, y and z and leaves same remainder = HCF (𝑥−𝑦, 𝑦−𝑧, 𝑧−𝑥)

Greatest number which divides x, y and z and leaves remainder a, b, c = HCF(x  a, y  b, z  c)
LCM Remainder Problems
4 Types
Smallest Number Divisible by X, Y and Z? LCM Type 1 (Simple)
Let’s start with a simple example: 6, 9, and 12
Smallest/Largest Number of N Digits Divisible by X, Y, Z? LCM Type 2 (Multiples of LCM)
Smallest/Largest number of 3 digits divisible by 6, 9, 12
Smallest Number when Divided by X, Y and Z Leaves Same Remainder R (Given)?LCM Type 3 (Same Remainder)
Number divisible by 6, 9, and 12 leaves remainder 2
Smallest Number when Divided by X, Y and Z Leaves Remainder a, B, C?LCM Type 4 (Different Remainder)
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
LCM Problems – 4 Types SummaryUnderstand and Remember

Smallest number divisible by x, y and z = LCM(x, y, z)

Smallest number of n digits divisible by x, y and z = Multiple of LCM(x, y, z)

Smallest number when divided by x, y and z leaves same remainder r = LCM(x, y, z) + r

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

Variations of LCM (Understand)

Smallest/Largest number of n digits when divided by x, y and z leaves same remainder r = Multiple of LCM(x, y, z) + r

Smallest/Largest number of n digits when divided by x, y and z leaves remainder a, b, c = Multiple of LCM(x, y, z) – d (xa = yb = zc = common difference d)
Example  1
Find the greatest number of 5digits which on being divided by 9, 12, 24 and 45 leaves 3, 6, 18 and 39 as remainders respectively.
Example  2
Find the smallest number which, on being added 23 to it, is exactly divisible by 32, 36, 48 and 96.
Example – 3 (Advanced)

When dividing a number by 12, 15 or 48 there will always be a remainder of 10. If the number is the least possible, how many divisors does the number have?

Number of divisors of (p, q primes) is Application of Combination
Generalization – Chinese Remainder Theorem Next Class!!
Find the smallest number which when divided by 7, 9, and 11 produces 1, 2, and 3 as reminders

7 – 1 = 6

9 – 2 = 7

11 – 3 = 8
But

7 – 2 × 1 = 5

9 – 2 × 2 = 5

11 – 2 × 3 = 5