Govt Exams
This question asks us to find the Highest Common Factor (HCF) of two numbers using prime factorization or the Euclidean algorithm.
Express 56 as a product of prime numbers.
Express 72 as a product of prime numbers.
The HCF is the product of common prime factors with their lowest powers.
The HCF of 56 and 72 is 8, making the correct answer (B).
Prime factorization: 12 = 2² × 3, 18 = 2 × 3². LCM = 2² × 3² = 4 × 9 = 36 (taking highest powers of all prime factors).
Using prime factorization: 48 = 2⁴ × 3, 64 = 2⁶. HCF = 2⁴ = 16 (taking lowest powers of common prime factors).
If a number is divisible by both 3 and 5, and 3 and 5 are coprime (HCF=1), then the number must be divisible by their product: 3×5 = 15
Factors of 48: 1,2,3,4,6,8,12,16,24,48.
Factors of 64: 1,2,4,8,16,32,64.
Common factors: 1,2,4,8,16. HCF = 16
Check: 21=3×7 (not prime), 23 is only divisible by 1 and 23 (prime), 25=5×5 (not prime), 27=3×9 (not prime).
So 23 is the smallest prime greater than 20.
Sum of first n natural numbers = n(n+1)/2.
Here n=10, so sum = 10(11)/2 = 110/2 = 55