In how many different ways can 840 be expressed as a product of two positive factors?

Question

In how many different ways can 840 be expressed as a product of two positive factors?

1. A. 20
2. B. 16 (Correct answer)
3. C. 10
4. D. 12

Correct Answer

Option B — 16

Detailed Solution & Explanation

The correct answer is 16.

Key Points

• Factorise: 840 = 2³ × 3 × 5 × 7.
• The number of divisors = (3 + 1)(1 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 × 2 = 32.
• Each unordered pair of factors uses up two divisors, so the number of ways = 32 ÷ 2 = 16.

Additional Information

• Halving is valid here because 840 is not a perfect square, so no divisor pairs with itself and the 32 divisors split cleanly into 16 pairs.
• For a perfect square the count would be (d + 1)/2, since the square root pairs with itself.
• Sample pairs: 1 × 840, 2 × 420, 3 × 280, 4 × 210, 5 × 168, 6 × 140, 7 × 120, 8 × 105, and so on to 20 × 42, 21 × 40, 24 × 35, 28 × 30.

*Ministry of Papers worked solution. The Commission has not yet published the official key for this paper; this answer will be reconciled with it on release.*