HCF of 3240, 3600 and a third number is 36, and their LCM is 24 × 35 × 52 ×…

Concept

For any set of numbers expressed in prime-factor form:

• the HCF takes the lowest power of each prime that is common to every number, and

• the LCM takes the highest power of each prime that appears in any number.

So an unknown number can be reconstructed prime by prime: each prime's exponent in the unknown is pinned by the HCF (a floor) and the LCM (a ceiling), together with what the known numbers already contribute.

Application

1. Factorise the known numbers and the HCF.

   • 3240 = 23 × 34 × 51

   • 3600 = 24 × 32 × 52

   • HCF = 36 = 22 × 32

   • LCM = 24 × 35 × 52 × 72

2. Find the exponent of each prime in the third number X.

   • Prime 2: known powers are 23 and 24; their minimum is already 23. For the HCF to be 22, X must pull the floor down to 22, so X has 22.

   • Prime 3: known powers are 34 and 32. The HCF needs the minimum to be 32 (already satisfied), while the LCM needs the maximum to be 35. Only X can raise it, so X has 35.

   • Prime 5: the HCF (36) has no factor of 5, so the minimum power of 5 across all three must be 0. Hence X has 50 — no factor of 5.

   • Prime 7: neither known number has a 7, but the LCM needs 72. X must supply it, so X has 72.

3. Assemble: X = 22 × 35 × 72.

Cross-check

Recombine all three numbers with X = 22 × 35 × 72:

• HCF takes the lowest power of each common prime: 22 (from min of 3, 4, 2) and 32 (from min of 4, 2, 5) = 22 × 32 = 36. ✓

• LCM takes the highest power of every prime: 24 × 35 × 52 × 72. ✓

Both conditions hold, confirming X = 22 × 35 × 72.