Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers:
(i) 26 and 91,
(ii) 510 and 92,
(iii) 336 and 54.
(i) LCM = 182; HCF = 13
(ii) LCM = 23460; HCF = 2
(iii) LCM = 3024; HCF = 6
To find the HCF and LCM using prime factorisation, we first write each number as a product of prime numbers. We test divisibility by primes in the correct order (2, 3, 5, 7, …), and once a number is not divisible by a smaller prime, we do not check that prime again for the same number.
HCF (Highest Common Factor) is the product of all common prime factors with the smallest power.
LCM (Least Common Multiple) is the product of all prime factors that appear in any of the numbers, using the highest power.
After that, we verify:
\( \text{LCM} \times \text{HCF} = \text{product of the two numbers} \)
(i) 26 and 91
Prime factorisation of 26:
26 is even, so it is divisible by 2.
\( 26 = 2 \times 13 \)
13 is prime.
Prime factorisation of 91:
91 is not divisible by 2.
Sum of digits = 10, not divisible by 3, so 91 is not divisible by 3.
Last digit is not 0 or 5, so not divisible by 5.
Check next prime 7: \(91 \div 7 = 13\).
So, \( 91 = 7 \times 13 \).
So we have:
\(26 = 2 \times 13\)
\(91 = 7 \times 13\)
HCF
Common prime = 13.
So, \( \text{HCF} = 13 \).
LCM
All primes used: 2, 7, 13
\( \text{LCM} = 2 \times 7 \times 13 = 182 \)
Verification
\( \text{LCM} \times \text{HCF} = \text{product of the two numbers} \)
\(182 \times 13 = 2366\)
\(26 \times 91 = 2366\)
Hence verified.
(ii) 510 and 92
Prime factorisation of 510:
510 is even ⇒ divide by 2.
\(510 = 2 \times 255\)
255 ends in 5 ⇒ divisible by 5.
\(255 = 5 \times 51\)
51 digit sum = 6 ⇒ divisible by 3.
\(51 = 3 \times 17\)
So, \(510 = 2 \times 5 \times 3 \times 17\)
Prime factorisation of 92:
92 is even ⇒ divisible by 2.
\(92 = 2 \times 46\)
46 is even again.
\(46 = 2 \times 23\)
23 is prime.
So, \(92 = 2^2 \times 23\)
So we have:
\(510 = 2^1 \times 3^1 \times 5^1 \times 17\)
\(92 = 2^2 \times 23\)
HCF
Common prime = 2.
Lowest power = \(2^1\).
So, HCF = 2.
LCM
All primes: 2, 3, 5, 17, 23
\( \text{LCM} = 2^2 \times 3 \times 5 \times 17 \times 23 = 23460 \)
Verification
\( \text{LCM} \times \text{HCF} = \text{product of the two numbers} \)
\(23460 \times 2 = 46920\)
\(510 \times 92 = 46920\)
Hence verified.
(iii) 336 and 54
Prime factorisation of 336:
336 is even:
\(336 = 2 \times 168\)
\(168 = 2 \times 84\)
\(84 = 2 \times 42\)
\(42 = 2 \times 21\)
21 is not even ⇒ check 3.
Digit sum = 3 ⇒ divisible by 3.
\(21 = 3 \times 7\)
So, \(336 = 2^4 \times 3 \times 7\)
Prime factorisation of 54:
54 is even:
\(54 = 2 \times 27\)
27 is not divisible by 2.
Digit sum = 9 ⇒ divisible by 3.
\(27 = 3^3\)
So, \(54 = 2 \times 3^3\)
So we have:
\(336 = 2^4 \times 3^1 \times 7\)
\(54 = 2^1 \times 3^3\)
HCF
Common primes = 2, 3
For 2: smallest power = 1
For 3: smallest power = 1
\( \text{HCF} = 2 \times 3 = 6 \)
LCM
Highest powers:
2 → \(2^4\)
3 → \(3^3\)
7 → \(7^1\)
\( \text{LCM} = 2^4 \times 3^3 \times 7 = 3024 \)
Verification
\( \text{LCM} \times \text{HCF} = \text{product of the two numbers} \)
\(3024 \times 6 = 18144\)
\(336 \times 54 = 18144\)
Hence verified.