Which of the following has the most number of divisors?
2024
Which of the following has the most number of divisors?
Answer: C. 176 — Concept: The number of divisors of a number can be found from its prime factorisation — write the number as a product of primes raised to powers, then…
- A.
99
- B.
123
- C.
176
- D.
987
Attempted by 5 students.
Show answer & explanation
Correct answer: C
Concept: The number of divisors of a number can be found from its prime factorisation — write the number as a product of primes raised to powers, then multiply together (each exponent + 1). For example, if a number factorises as p2 × q3 for distinct primes p and q, it has (2 + 1) × (3 + 1) = 12 divisors.
Application: factorise each of the four given numbers and apply the formula.
Number | Prime factorisation | Divisor count |
|---|---|---|
99 | 32 × 11 | (2 + 1)(1 + 1) = 6 |
123 | 3 × 41 | (1 + 1)(1 + 1) = 4 |
176 | 24 × 11 | (4 + 1)(1 + 1) = 10 |
987 | 3 × 7 × 47 | (1 + 1)(1 + 1)(1 + 1) = 8 |
Cross-check by direct listing: the divisors of 176 are 1, 2, 4, 8, 16, 11, 22, 44, 88 and 176 — 10 divisors in total, confirming the formula above.
So among these four numbers, 176 has the most divisors (10), as the divisor-count formula applied to each factorisation in the table above shows.