12 can be written as product of 3 numbers in 18 ways like (1,2,6), (1,3,4)…
2025
12 can be written as product of 3 numbers in 18 ways like (1,2,6), (1,3,4) etc. In how many ways can 3600 can be written like this -
- A.
441
- B.
540
- C.
84
- D.
2100
Attempted by 496 students.
Show answer & explanation
Correct answer: B
Answer: 540
Step 1 — Prime factorize the number:
3600 = 2^4 · 3^2 · 5^2
Step 2 — Distribute exponents among three factors (ordered triples):
For prime 2 (exponent 4): number of nonnegative integer solutions to a+b+c = 4 is C(4+3-1,3-1) = C(6,2) = 15.
For prime 3 (exponent 2): number of solutions is C(2+3-1,3-1) = C(4,2) = 6.
For prime 5 (exponent 2): number of solutions is C(2+3-1,3-1) = C(4,2) = 6.
Step 3 — Multiply the counts for each prime (independent choices):
Total ways = 15 × 6 × 6 = 540
Note: This counts ordered triples (a,b,c). If unordered triples are required, additional division or casework is needed.