# Find the two integers: visual puzzle and solution | SwaVid

Two positive consecutive integers multiply to 156. What are the integers? See the complete step-by-step solution.

Canonical: https://www.swavid.com/puzzles/consecutive-product

Source: https://www.swavid.com/puzzles/consecutive-product

# Find the two integers

## 12 and 13.

### Estimate with a square root

### Try the two numbers on either side

### Confirm with algebra

SwaVid visual puzzle

Two positive consecutive integers multiply to 156. What are the integers?

Detailed solution

Consecutive integers multiply to slightly more than a perfect square, so start from the square root.

156 sits between 144 and 169.

The product of consecutive integers lies between their squares.

Solving n(n + 1) = 156 gives n² + n − 156 = 0, which factors as (n − 12)(n + 13). Only the positive root counts.

Halving 156 to get 78 is a factor pair (2 × 78) but not a consecutive one.

- 1 Estimate with a square root 156 sits between 144 and 169. 12² = 144, 13² = 169
- 2 Try the two numbers on either side The product of consecutive integers lies between their squares. 12 × 13 = 156 ✓
- 3 Confirm with algebra Solving n(n + 1) = 156 gives n² + n − 156 = 0, which factors as (n − 12)(n + 13). Only the positive root counts. n = 12

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