A gardener organizes plants into rows to create a square formation but discovers that 15 plants are not included. If the total number of plants is 3984, then what is the number of plants in each row?

Correct — C. The plants that fit form a perfect square: as many rows as plants in each row.

Plants in the square: 3984 − 15 = 3969
Bracket it: 60² = 3600 and 70² = 4900, so the root lies in the 60s
Last digit 9: the root ends in 3 or 7, so try 63

Square it: (60 + 3)² = 3600 + 360 + 9 = 3969
Plants in each row: 63 → option (c)

Why the others are wrong

• (a)
  43 — 43² = 1849, and 1849 + 15 = 1864 plants, less than half the 3984 the gardener has.
• (b)
  45 — 45² = 2025, so this square plus the 15 left over is only 2040 plants, not 3984.
• (d)
  65 — 65² = 4225, more than the 3984 plants in total. A square that size cannot be formed at all, let alone with 15 to spare.

A square formation has as many rows as plants in each row, so a formation of n rows uses n² plants. Whatever does not fit is left over: total = n² + leftover.

So take the leftover away first, then find the square root. Prime factors confirm it: 3969 = 3⁴ × 7², so √3969 = 3² × 7 = 63.

Key facts

• A square formation of n rows holds n² items, so total = n² + leftover.
• 3969 = 3⁴ × 7², so √3969 = 63.
• A perfect square never ends in 2, 3, 7 or 8.
• A perfect square ending in 9 has a square root ending in 3 or 7.

Common traps

• Adding the 15 instead of subtracting it: 3984 + 15 = 3999 is not a perfect square, since 63² = 3969 and 64² = 4096.
• Taking √3984 ≈ 63.1 as a sign something is wrong. The 15 plants left over are exactly why 3984 is not a perfect square.

No directly related past PYQ was found.