# Cracking LCM & HCF Remainder Problems: 8 Simple Formulas Explained YouTube Lecture Handouts

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## Recap

Method:

• Prime Factorization

• Divisibility tests

Problem Keywords

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

• HCF: Maximum number, most amount, longest duration etc.

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)

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 (x-a = y-b = z-c = common difference d)

## Example - 1

Find the greatest number of 5-digits 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