Question

Can two numbers have 15 as their HCF and 175 as their LCM? Give reason.

Answer 1

Given: Two numbers with HCF = 15 and LCM = 175.

Step-wise calculation:

1. For any two positive integers a and b, a × b = HCF(a, b) × LCM(a, b).

Hence, product a × b

= 15 × 175

= 2625

2. Let the two numbers be 15 m and 15 n where gcd(m, n) = 1 since 15 is their HCF.

Then LCM(15m, 15n) = 15 × m × n because m and n are coprime. 

So, 15 × m × n = 175

⇒ m × n = `175/15`

= `35/3`, which is not an integer. 

Therefore, no such integer m, n exist.

3. Prime-factor check (simpler): 15 = 3 × 5 must divide the LCM.

But 175 = 5² × 7 has no factor 3, so 15 cannot divide 175. 

Thus, 15 cannot be HCF when LCM = 175.

No. Two integers cannot have 15 as their HCF and 175 as their LCM, because 15 does not divide 175 equivalently, the required coprime factors m × n would not be an integer.