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Consider N = 15!. Which of the following statements are correct?
- The product of all divisors of N is (15!)²⁰¹⁶.
- The sum of all odd divisors of N equals (7³ - 1)(5⁴ - 1)(7³ - 1) - 7 / 2.
- The product of all divisors of N that are not divisible by 5 equals (2¹¹ * 3⁶ * 7² * 11 * 13)⁵⁰⁴.
- The sum of all perfect-square divisors of N equals ((2¹² - 1)/3) * ((3⁸ - 1)/8) * 25.
Correct answer: The product of all divisors of N is (15!)²⁰¹⁶.
Solution
15! = 2¹¹ * 3⁶ * 5³ * 7² * 11 * 13. d(15!) = 12*7*4*3*2*2 = 4032. A: Product = N^(d/2) = (15!)²⁰¹⁶. CORRECT. B: Sum of odd divisors excludes powers of 2 and 5. The formula given has 7³ repeated (typo) and wrong structure — INCORRECT. C: Divisors not divisible by 5 come from 2¹¹*3⁶*7²*11*13; their count = 12*7*3*2*2 = 1008; product = (2¹¹*3⁶*7²*11*13)⁵⁰⁴. CORRECT. D: Perfect-square divisors: even exponents only. From 2¹¹ get choices 2⁰,2²,...,2¹⁰ (6); from 3⁶ get 3⁰,...,3⁶ (4); from 5³ get 5⁰,5² (2); from 7² get 7⁰,7² (2); 11⁰,13⁰ each (1). Sum = ((2¹²-1)/3) * ((3⁸-1)/8) * 26 * 50 * 1 * 1. The factor 26*50 = 1300, not 25 — INCORRECT.
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