## LCM & HCF

Lowest Common Multiple (LCM)

LCM is the least dividend which is exactly divisible by the given numbers.

LCM of two or more numbers is the product of the highest powers of all prime factors of the given numbers.

**Question 1: Find the LCM of 2**

^{3}×3^{4}×5^{2}, 2^{2}×3^{3}and 2^{2}×3^{3}×5^{2}×7^{2}(a) 2

^{3}×3

^{4}×5

^{2}×7

^{2}

(b) 2

^{2}×3

^{3}×5

^{1}×7

^{1}

(c) 2×3×5×7

(d) 2

^{3}×3^{3}×5^{2}×7^{2}^{}

^{3}×3

^{4}×5

^{2}×7

^{2}

^{}

**Highest Common Factor (HCF)**

HCF of two or more numbers is the greatest number (divisor) that divides all the given numbers exactly. HCF is also known as the GREATEST COMMON DIVISOR (GCD).

For a given two numbers a and b,

HCF × LCM = a × b

**Question 2: Find the HCF of 120, 150,180.**

(a) 30

(b) 60

(c) 50

(d) 10

Answer: (a) 30

[Hint: Take the least powers of common terms ]

**Question 3: The HCF of two numbers is 11 and the LCM is 7700. If one of the numbers is 275, then find the other number.**

(a) 550

(b) 308

(c) 77

(d) 7700

Answer: (b) 308

11 × 7700 = 275 × a

a = (7700 × 11) / 275 = 308

**Question 4: The least square number exactly divisible by 8, 12, 15 and 20 is:**

(a) 900

(b) 1200

(c) 3600

(d) 14400

Answer: (c) 3600

LCM of 8, 12, 15 and 20 is = 2

^{3}, 2

^{2}×3, 3×5, 2

^{2}×5 = 2

^{3}× 3 × 5 = 120

120 is not a perfect square.

Making it a perfect square number = 120 × 3 ×2 × 5 = 3600

**Question 5: Six bells rings at intervals of 2, 3, 6, 10 and 12 seconds respectively. How many times the bells will ring together in 1 hour?**

(a) 2 times

(b) 20 times

(c) 30 times

(d) 120 times

Answer: (c) 30 times

L.C.M of 2, 4, 6, 8, 10 and 12 is 120.

The bells will ring together after 120 seconds i.e. 2 minutes.

In 1 hour i.e. 60 mins it will ring together 60/2 = 30 times