Difficulty: Hard
Correct Answer: 176
Explanation:
Introduction / Context:
This question tests your understanding of how to count the number of divisors of a number using prime factorization. When a number is expressed as a product of prime powers, there is a formula that gives the count of its positive divisors. We apply this to each option and then compare the results.
Given Data / Assumptions:
Concept / Approach:
If n is written as:
n = p1^a * p2^b * p3^c * ...
then the total number of positive divisors of n is:
(a + 1)(b + 1)(c + 1)...
We factor each option, compute the divisor count using this formula, and then choose the largest.
Step-by-Step Solution:
Step 1: Factorize 99.99 = 9 * 11 = 3^2 * 11^1.Divisors of 99 = (2 + 1)(1 + 1) = 3 * 2 = 6.Step 2: Factorize 101.101 is prime.Divisors of 101 = 2 (1 and 101).Step 3: Factorize 176.176 = 16 * 11 = 2^4 * 11^1.Divisors of 176 = (4 + 1)(1 + 1) = 5 * 2 = 10.Step 4: Factorize 182.182 = 2 * 91 = 2^1 * 7^1 * 13^1.Divisors of 182 = (1 + 1)(1 + 1)(1 + 1) = 2 * 2 * 2 = 8.Step 5: Compare divisor counts: 99 has 6, 101 has 2, 176 has 10, and 182 has 8. The maximum is 10 for the number 176.
Verification / Alternative check:
You can confirm by listing divisors of 176: 1, 2, 4, 8, 16, 11, 22, 44, 88, 176. There are exactly 10, which matches the formula. The other numbers have fewer divisors when listed explicitly.
Why Other Options Are Wrong:
99: Only 6 divisors, fewer than 176.101: Prime, so it has only 2 divisors.182: Has 8 divisors, again fewer than 176.120 (extra option): It has many divisors, but it is not among the original four choices we are asked to compare.
Common Pitfalls:
Assuming that the largest number must have the most divisors.Miscomputing prime factorization and exponents, which leads to incorrect divisor counts.Forgetting to add 1 to each exponent before multiplying them to get the total number of divisors.
Final Answer:
176
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