the lcm of two numbers is 7700, and their hcf is 11. if one of these numbers is 275, what is the other number?

The other number is 308. To solve this, recall that for any two numbers aaa and bbb, the relationship is gcd⁡(a,b)×lcm⁡(a,b)=a×b\gcd(a,b)\times \operatorname{lcm}(a,b)=a\times bgcd(a,b)×lcm(a,b)=a×b. Here, gcd⁡=11\gcd =11gcd=11, lcm⁡=7700\operatorname{lcm}=7700lcm=7700, and one number a=275a=275a=275, so b=11×7700275b=\frac{11\times 7700}{275}b=27511×7700​.

First, simplify: 7700÷275=287700\div 275=287700÷275=28 (since 275×28=7700275\times 28=7700275×28=7700). Then, b=11×28=308b=11\times 28=308b=11×28=308.

Verify: Both 275 (11×2511\times 2511×25) and 308 (11×2811\times 2811×28) share gcd⁡=11\gcd =11gcd=11, and lcm⁡(25,28)=700\operatorname{lcm}(25,28)=700lcm(25,28)=700 (as 25 and 28 are coprime), so lcm⁡=11×700=7700\operatorname{lcm}=11\times 700=7700lcm=11×700=7700.

Step-by-Step Breakdown

1. Write the formula: 11×7700=275×b11\times 7700=275\times b11×7700=275×b.
2. Compute product: 11×7700=8470011\times 7700=8470011×7700=84700.
3. Solve: b=84700÷275=308b=84700\div 275=308b=84700÷275=308.

Quick Facts

• Prime factors : 275=11×52275=11\times 5^2275=11×52, 308=11×22×7308=11\times 2^2\times 7308=11×22×7.

• Why it works : Numbers are HCF times coprime parts; product of parts gives LCM/HCF ratio.

TL;DR: The other number is 308.