What is the smallest number that must be added to 2378 to make it a perfect square?

Correct — A. Find the perfect squares on either side of 2378.

48² = 2304, which is below 2378
49² = 2401, the next square above

Number to add = 2401 − 2378 = 23
Check: 2378 + 23 = 2401 = 49² → option (a)

Why the others are wrong

• (b)
  24 — 2378 + 24 = 2402, one more than 49² = 2401. No perfect square ends in 2, and the next square, 50² = 2500, is much higher.
• (c)
  25 — 2378 + 25 = 2403, which ends in 3. No perfect square ends in 2, 3, 7 or 8, so 2403 cannot be one.
• (d)
  26 — 2378 + 26 = 2404 ends in 4, but it lies between 49² = 2401 and 50² = 2500, so it is not a perfect square.

To make a number a perfect square by adding, climb to the next square above it. To do it by subtracting, drop to the square just below.

So first trap the number between two consecutive squares. 48² = 2304 and 49² = 2401 trap 2378, so the next square is 49².

Smallest number to add = next square − the number: 2401 − 2378 = 23.

Key facts

• 48² = 2304, 49² = 2401 and 50² = 2500.
• Smallest number to add = the next square above − the number.
• Smallest number to subtract = the number − the square below (2378 − 2304 = 74).
• No perfect square ends in 2, 3, 7 or 8.

Common traps

• Mixing up the directions. Adding climbs to 49² = 2401 (add 23), while subtracting drops to 48² = 2304 (subtract 74).
• Trusting the last digit alone. An ending of 0, 1, 4, 5, 6 or 9 does not prove a square: 2404 ends in 4 and is not one.

No directly related past PYQ was found.