Find the HCF and the LCM of $306$ and $657$, and verify that HCF $\times$ LCM $=$ product of the two numbers.

Step-by-step solution

1. Write each number as a product of primes. $306=2\times 3^2\times 17$ and $657=3^2\times 73$.
2. The HCF is the product of the common prime factors, each taken with the smallest power it appears with. Only $3^2$ is common, so HCF $=3^2=9$.
3. The LCM is the product of all prime factors, each taken with the greatest power: LCM $=2\times 3^2\times 17\times 73=22338$.
4. Now check the property. HCF $\times$ LCM $=9\times 22338=201042$.
5. And the product of the two numbers is $306\times 657=201042$. The two agree, so the result is verified.

Concept tested: Relationship between HCF and LCM

Related questions

• 1 Without performing long division, state whether 13/3125 has a terminating or a non-terminating repeating decimal expansion. Write the expansion if it terminates.
• 2 Explain why 7 × 11 × 13 + 13 is a composite number.
• 3 Determine whether the pair of equations 5x - 2y = 7 and 10x - 4y = 20 are consistent, inconsistent, or represent the same straight line. Justify your answer.
• 4 Check whether 6^n can end with the digit 0 for any natural number n.
• 5 Find the zeros of the quadratic polynomial x^2-5x+6.
• 6 Solve the pair of equations 2x + 3y = 7 and x - y = 1 by the method of substitution.