Page 67 - ICSE Math 6

Now, the multiples of 120 are 240, 360, 480, 600, ...,
9,840, 9,960, 10,080, ..., etc.

Hence, the greatest 4-digit number which is exactly divisible by 8, 10 and 12 is 9,960.
Alternatively,

The greatest 4-digit number is 9,999.
The LCM of 8, 10 and 12 is 2 × 2 × 2 × 5 × 3 = 120.

9,999 leaves the remainder 39 when divided by 120
Hence, the required number is 9,999 – 39 = 9,960.

Relation between HCF and LCM of Two Numbers

For any two numbers, HCF × LCM = Product of the two numbers.
From the above relation, we can deduce the following:
Product of the two numbers
• HCF = LCM

Product of the two numbers
• LCM = HCF

HCF × LCM
• = Other number
One of the numbers

• As the HCF of two co-prime numbers is 1, therefore the LCM of two co-primes is the product of
the co-primes.
Example 35: Verify the relation between HCF and LCM of 75 and 135.

Solution: Prime factorize 75 and 135.
75 = 3 × 5 × 5 and 135 = 3 × 3 × 3 × 5
HCF of 75 and 135 = 3 × 5 = 15
2
3
LCM of 75 and 135 = 3 × 5 = 27 × 25 = 675
HCF × LCM = 15 × 675 = 10,125
Product of 75 × 135 = 10,125
Thus, HCF × LCM = Product of the two numbers

EXERCISE 4.6

1. Find the LCM of the following numbers using the common multiple method.

(a) 5 and 9 (b) 12 and 16 (c) 36, 39 and 120
(d) 132 and 264 (e) 144, 612 and 204 (f) 32, 64, 720 and 120
2. Find the LCM of the following numbers using the prime factorization method.
(a) 4, 6 and 10 (b) 16, 25 and 35 (c) 112 and 222
(d) 24, 25 and 99 (e) 45, 75 and 100 (f) 33, 66 and 88

3. Find the LCM of the following numbers using the division method.
(a) 3 and 27 (b) 5, 25 and 75 (c) 20, 40 and 50
(d) 12, 16 and 48 (e) 42, 56 and 80 (f) 36, 54, 72 and 27

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