Permutations & Combinations
Factorials and divisors
Grade None
Question:
<p>If \(10! = 2^p \cdot 3^q \cdot 5^r \cdot 7^s\), then</p>
<p>\(2q = p\)</p>
<p>\(pqrs = 64\)</p>
<p>number of divisors of \(10!\) is 280</p>
<p>number of ways of putting \(10!\) as a product of two natural numbers is 135</p>
Step-by-Step Solution
Key Concept: Use Legendre's formula to find the highest power of each prime dividing n!: the exponent of prime p in n! is ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + ...
<p><strong>Step 1:</strong> Apply Legendre's formula for p = 2:<br/>⌊10/2⌋ + ⌊10/4⌋ + ⌊10/8⌋ = 5 + 2 + 1 = 8<br/>So p = 8</p><p><strong>Step 2:</strong> Apply Legendre's formula for p = 3:<br/>⌊10/3⌋ + ⌊10/9⌋ = 3 + 1 = 4<br/>So q = 4</p><p><strong>Step 3:</strong> Apply Legendre's formula for p = 5:<br/>⌊10/5⌋ + ⌊10/25⌋ = 2 + 0 = 2<br/>So r = 2</p><p><strong>Step 4:</strong> Apply Legendre's formula for p = 7:<br/>⌊10/7⌋ + ⌊10/49⌋ = 1 + 0 = 1<br/>So s = 1</p><p><strong>Verification:</strong> 10! = 3,628,800 = 2⁸ · 3⁴ · 5² · 7¹ ✓</p><p>∴ <strong>p = 8, q = 4, r = 2, s = 1</strong></p>
Correct Answer: A