Key facts

  • A whole-number perfect square has paired prime factors, while a whole-number perfect cube has prime factors grouped in threes;
  • Unit digits give quick rejection clues: a perfect square cannot end in 2, 3, 7 or 8, but a valid unit digit alone never proves that a number is a squa...
  • For two numbers, product of the numbers = HCF × LCM; this relation is useful only when exactly two positive integers are involved.
  • LCM answers “when will events meet again?” and HCF answers “what is the greatest equal size?” in most word problems.
  • Factorisation in CET is usually based on common factors, identities, difference of squares and middle-term splitting, not advanced algebra.

Key Points at a Glance

  1. 1

    A whole-number perfect square has paired prime factors, while a whole-number perfect cube has prime factors grouped in threes; this is the safest recognition method.

  2. 2

    Unit digits give quick rejection clues: a perfect square cannot end in 2, 3, 7 or 8, but a valid unit digit alone never proves that a number is a square.

  3. 3

    For two numbers, product of the numbers = HCF × LCM; this relation is useful only when exactly two positive integers are involved.

  4. 4

    LCM answers “when will events meet again?” and HCF answers “what is the greatest equal size?” in most word problems.

  5. 5

    Factorisation in CET is usually based on common factors, identities, difference of squares and middle-term splitting, not advanced algebra.

  6. 6

    A quadratic equation has standard form ax² + bx + c = 0, where a is not 0; roots are found by factorisation or by the formula when factors are not obvious.

  7. 7

    Logarithm means the power needed to obtain a number from a base; its laws work for products, quotients and powers, not for ordinary addition.

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Whole-number squares, square roots, cubes and cube roots

A square is obtained when a number is multiplied by itself. Thus 17² = 17 × 17 = 289. A square root reverses this operation: √289 = 17. For CET questions, “whole-number square root” usually means the given number is a perfect square and the root is a whole number. If the number is not a perfect square, the exact square root may be irrational, but the question may still ask for an estimate.

A cube is obtained when a number is multiplied by itself three times. Thus 6³ = 6 × 6 × 6 = 216. A cube root reverses this operation: ∛216 = 6. Here also, a perfect cube has a whole-number cube root.

Prime factorisation gives the most reliable test. A perfect square must have every prime factor to an even power. For example, 784 = 2⁴ × 7², so √784 = 2² × 7 = 28. A perfect cube must have every prime factor to a power divisible by 3. For example, 1728 = 2⁶ × 3³, so ∛1728 = 2² × 3 = 12.

This method also prevents false assumptions from unit digits. A number ending in 6 may be a square, but it is not automatically one. For example, 36 is a square, but 46 is not. Always combine a quick clue with factorisation, nearby square comparison or the answer options.

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