Question

If p, q, r and s are four positive integers such that pqrs = 1, what is the minimum value of (2 + p)(3 + q)(4 + r)(5 + s)?

• A. 1
• B. 12
• C. 120
• D. 240
• E. 360

Answer 1

Answer: (E)

To find the minimum value of \((2 + p)(3 + q)(4 + r)(5 + s)\), given that \(pqrs = 1\) and \(p, q, r, s\) are positive integers, we must work through the problem systematically.

1. Since \(p\), \(q\), \(r\), and \(s\) are positive integers whose product is 1, the only possibility is that each of them is equal to 1. That is, \(p = q = r = s = 1\).
2. Substitute these values into the expression \((2 + p)(3 + q)(4 + r)(5 + s)\):
   • \(2 + p = 2 + 1 = 3\)
   • \(3 + q = 3 + 1 = 4\)
   • \(4 + r = 4 + 1 = 5\)
   • \(5 + s = 5 + 1 = 6\)
3. Now, calculate the product of these terms: 3 \times 4 \times 5 \times 6
   • \(3 \times 4 = 12\)
   • \(12 \times 5 = 60\)
   • \(60 \times 6 = 360\)

Therefore, the minimum value of \((2 + p)(3 + q)(4 + r)(5 + s)\) is 360.

Hence, the correct answer is 360.

Note: Since the only integers that multiply to 1 are 1 itself, the options of 12, 120, 240, which are smaller than 360, are not feasible given that each factor involves adding to these numbers, consequently increasing the product.