Question

Which of these numbers has the most number of divisors?

• A. 200
• B. 156
• C. 240
• D. 172

Answer 1

Answer:

C. 240

ANSWER: number 240 has the most number of divisors.

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To easily find the total number of divisors (factors), we can express each number by its prime factorization and use the formula:
$\text{Total Divisors} = (p_1 + 1) \times (p_2 + 1) \times \dots \times (p_n + 1)$
(where $p_1, p_2, \dots$ are the exponents of its prime factors).

1. Number: 200

• Prime Factorization: $200 = 2^3 \times 5^2$

• Number of Divisors: $(3 + 1) \times (2 + 1) = 4 \times 3 = \mathbf{12}$

2. Number: 156

• Prime Factorization: $156 = 2^2 \times 3^1 \times 13^1$

• Number of Divisors: $(2 + 1) \times (1 + 1) \times (1 + 1) = 3 \times 2 \times 2 = \mathbf{12}$

3. Number: 240

• Prime Factorization: $240 = 2^4 \times 3^1 \times 5^1$

• Number of Divisors: $(4 + 1) \times (1 + 1) \times (1 + 1) = 5 \times 2 \times 2 = \mathbf{20}$

4. Number: 172

• Prime Factorization: $172 = 2^2 \times 43^1$

• Number of Divisors: $(2 + 1) \times (1 + 1) = 3 \times 2 = \mathbf{6}$

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Number	Prime Factorization	Total Divisors
200	$2^3 \times 5^2$	12
156	$2^2 \times 3^1 \times 13^1$	12
240	$2^4 \times 3^1 \times 5^1$	20
172	$2^2 \times 43^1$	6