CAT Quant · Chapter Test
Perfect Squares — 25-Question Test
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Formula Reference Sheet
This chapter
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⌋ |
CAT reference
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