The statement below is followed by four options. Each option is related to the item in the statement, but only one is always true. Select the option that always holds and makes the statement true.
"The square of a natural number always has ______."
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → An odd number of factors
The answer must hold for every square of a natural number, without exception. An option that is present in most cases but can be missing in even one case is not always true.
Option (1) An even number of factors — 9 has three factors (1, 3, 9), an odd number.
Option (2) An odd number of factors — factors come in pairs (a, n/a); in a square one pair is a factor times itself (3 × 3 in 9, 4 × 4 in 16), which is counted once, so the count of factors is always odd (1 has one factor, 16 has five), so this is always true.
Option (3) A last digit of 1, 4 or 9 — 25 = 5 × 5 ends in 5 and 36 = 6 × 6 ends in 6.
Option (4) Exactly three factors — 16 has five factors (1, 2, 4, 8, 16).