Square Properties
Two fast filters reject most non-squares instantly. First, the unit digit: a perfect square can only end in 0, 1, 4, 5, 6 or 9. So anything ending in 2, 3, 7 or 8 is never a square — no work needed. Second, the digital root (the single digit you reach by repeatedly adding the digits): for a perfect square it must be 1, 4, 7 or 9. A number whose digital root is 2, 3, 5, 6 or 8 cannot be a square. Both tests are necessary but not sufficient — passing them does not prove a number IS a square, it only fails to reject it. A third structural fact CAT loves: between n² and (n+1)² there are exactly 2n non-square integers, because consecutive squares differ by 2n+1. Also useful: every odd square is 1 more than a multiple of 8, and the sum of the first n odd numbers equals n².
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.
Formula Reference Sheet
Identity & gap rules
| Perfect square definition | N = k² for some integer k ≥ 0 |
|---|---|
| Possible unit digits | last digit ∈ {0, 1, 4, 5, 6, 9} only |
| Digital root of a square | digital root ∈ {1, 4, 7, 9} only |
| Non-squares between n² and (n+1)² | (n+1)² − n² − 1 = 2n |
| Sum of first n odd numbers | 1 + 3 + 5 + … + (2n−1) = n² |
Factor-count power-tools
| Number of factors | if N = p^a · q^b · r^c then d(N) = (a+1)(b+1)(c+1) |
|---|---|
| Perfect-square test | N is a square ⇔ every prime exponent a, b, c … is even |
| Odd factor count | d(N) is odd ⇔ N is a perfect square |
| Smallest multiplier to a square | multiply by the product of primes with odd exponent |
| Squares ≤ M | count = ⌊√M⌋ |