Divisibility Rules; LCM and HCF

Divisibility Rules

2.1 Rules for Each Number

2.2 Examples

Check divisibility of 792 by 11:

• Odd position digits (from right): 7, 9, 2 → No wait

Actually: positions from LEFT: 7(pos 1), 9(pos 2), 2(pos 3)

• Odd position sum = 7 + 2 = 9

Even position sum = 9

• Difference = 9 - 9 = 0 → Divisible by 11 ✓

Check if 348 is divisible by 4:

• Last 2 digits = 48
• 48 ÷ 4 = 12 → Yes ✓

Check if 4563 is divisible by 3:

• Sum of digits = 4 + 5 + 6 + 3 = 18
• 18 ÷ 3 = 6 → Yes ✓

2.3 Divisibility Rule for 7 (Detailed)

• Take the number, double the last digit, subtract from the remaining number
• Repeat until you get a small number
• Example: 343 → 34 - (2×3) = 34 - 6 = 28 → 28 is divisible by 7 → 343 is divisible by 7 ✓

LCM and HCF

3.1 Key Definitions

HCF (Highest Common Factor) = GCD = Greatest Common Divisor

• The LARGEST number that divides all given numbers exactly
• Also called GCD (Greatest Common Divisor)

LCM (Least Common Multiple)

• The SMALLEST number that is exactly divisible by all given numbers

3.2 Key Relationship

HCF × LCM = Product of two numbers (valid for exactly 2 numbers only)

If HCF of two numbers is H and LCM is L and one number is a, then other = (H × L) / a

3.3 Methods to Find HCF

Prime Factorization Method

• Factorize each number into prime factors
• HCF = Product of LOWEST powers of common factors

Example: HCF(12, 18, 24)

• 12 = 2² × 3
• 18 = 2 × 3²
• 24 = 2³ × 3
• HCF = 2¹ × 3¹ = 6

Division Method (Euclid's Algorithm)

• Divide larger by smaller, remainder becomes divisor, previous divisor becomes dividend
• Continue until remainder = 0
• Last divisor = HCF

Example: HCF(48, 18)

• 48 ÷ 18 = 2 remainder 12
• 18 ÷ 12 = 1 remainder 6
• 12 ÷ 6 = 2 remainder 0
• HCF = 6

3.4 Methods to Find LCM

Prime Factorization Method

• LCM = Product of HIGHEST powers of all prime factors

Example: LCM(12, 18, 24)

• 12 = 2² × 3
• 18 = 2 × 3²
• 24 = 2³ × 3
• LCM = 2³ × 3² = 8 × 9 = 72

Division Method

• Write all numbers in a row
• Divide by smallest prime that divides at least one number
• Continue until all quotients are 1
• LCM = Product of all divisors used

3.5 Important Formulas

3.6 Word Problems — Key Tricks

When remainder is same and given:

If N leaves remainder r with a, b, c → N - r is divisible by HCF(a, b, c)

• Find HCF of (a, b, c) and add r

Example: Find the greatest number that divides 276, 690 and 1610 leaving remainder 6:

Subtract remainder: 270, 684, 1604

• HCF(270, 684, 1604) = HCF using difference method
• 690
• 276 = 414; 1610 - 690 = 920
• HCF(414, 920) = 46 → Greatest number = 46

When remainders are different:

• If N leaves different remainders r1, r2, r3 with a, b, c respectively:
  • (a - r1) = (b - r2) = (c - r3) = constant k
  • N = LCM(a, b, c) - k

LCM with common remainder:

• Result = LCM(a, b, c) + common remainder

HCF(42, 63, 105):

• 42 = 2 × 3 × 7
• 63 = 3² × 7
• 105 = 3 × 5 × 7
• HCF = 3 × 7 = 21

If HCF = 8, LCM should have factor of 8

• If four numbers are in ratio a:b:c:d with HCF = h → numbers = ah, bh, ch, dh
• Example: Four numbers in ratio 10:12:15:18 with HCF = 3 → Numbers = 30, 36, 45, 54