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AFCAT Numerical Ability · Squares, Cubes and Roots

The Least Number That Makes a Perfect Square or Cube

Find the least number to multiply or divide by for a perfect square or cube, the greatest square or cube that divides a number, and the least number to add or subtract for a perfect square.

All Squares, Cubes and Roots notes

Why this matters

Seven of the page's ten questions are HARD, the highest share of any page in the chapter. Almost all are one move, writing the number in prime factors, and marks go on reading divide as multiply, or on topping a cube's exponents up to the next even number instead of the next multiple of 3.

Concept 1 of 2: Least multiplier or divisor, by prime factors

Write the number in prime factors. A square needs every exponent even; a cube needs every exponent a multiple of 3. The least multiplier supplies the primes that are missing; the least divisor removes the primes left over.

Definition

  • Square: the primes with odd exponents are the leftovers. 50=2×5250 = 2 \times 5^2: multiply by 2 to get 100=102100 = 10^2, or divide by 2 to get 25=5225 = 5^2.
  • Cube: top each exponent up to the next multiple of 3, or cut it down to the one below. 1350=2×33×521350 = 2 \times 3^3 \times 5^2: multiply by 22×5=202^2 \times 5 = 20 to get 27000=30327000 = 30^3, or divide by 2×52=502 \times 5^2 = 50 to get 27=3327 = 3^3.
  • Greatest square or cube that divides n: keep the largest even part (square) or the largest multiple of 3 (cube) of each exponent. For 864=25×33864 = 2^5 \times 3^3 the greatest square factor is 24×32=1442^4 \times 3^2 = 144, and the greatest cube factor is 23×33=2162^3 \times 3^3 = 216.
  • The new root is read off the new exponents, by halving them or dividing them by 3.

What a perfect power needs

square: every exponent even,cube: every exponent a multiple of 3\text{square: every exponent even}, \quad \text{cube: every exponent a multiple of } 3

Worked example

Find the least number by which 2,000 must be multiplied, and the least by which it must be divided, to make a perfect cube.
Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Squares, Cubes and Roots · Making a Perfect Square or Cube

What is the smallest number by which 26244 must be divided to get a perfect cube?

Least means only the missing primes

72=23×3272 = 2^3 \times 3^2. Multiplying by 8 also gives a square (576), but the least multiplier is 2, giving 144.

A cube needs threes

40=23×540 = 2^3 \times 5. A cube needs two more 5s, so multiply by 25 to reach 1,000. Multiplying by 5 gives 23×522^3 \times 5^2, still not a cube.

Read the verb

Dividing removes the leftover primes; multiplying adds the missing ones. For a cube the two answers usually differ, so answer the one asked.
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Concept 2 of 2: Least number to add or subtract for a perfect square

Find the two consecutive squares on either side of the number. To add, go up to the square above; to subtract, go down to the square below.

Definition

  • Bracket nn between k2k^2 and (k+1)2(k + 1)^2. The least number to add is (k+1)2−n(k + 1)^2 - n; the least to subtract is n−k2n - k^2.
  • Find kk from a round square nearby and step: (k+1)2=k2+2k+1(k + 1)^2 = k^2 + 2k + 1, so after 502=250050^2 = 2500 come 2,601 and 2,704.
  • If n=k2n = k^2 is already a square, the least number to subtract for a smaller square is n−(k−1)2=2k−1n - (k - 1)^2 = 2k - 1.

Nearest squares

k2≤n<(k+1)2:add (k+1)2−n,subtract n−k2k^2 \le n < (k + 1)^2: \quad \text{add } (k + 1)^2 - n, \quad \text{subtract } n - k^2

Worked example

Find the least number to be added to 2,620, and the least number to be subtracted from it, to get a perfect square.
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Squares, Cubes and Roots · Making a Perfect Square or Cube

What is the smallest number that must be added to 1780 to make it a perfect square?

Which neighbour

Adding uses the square above; subtracting uses the square below. For 850 the answers are 50 (up to 900) and 9 (down to 841); swapping the neighbours mixes them up.

Step from a round square

Guessing the root by eye often lands one off. Start from a square you know, such as 302=90030^2 = 900, and step up or down by 2k+12k + 1.
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Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

  • Least multiplier or divisor, by prime factors

    What a perfect power needs

    square: every exponent even,cube: every exponent a multiple of 3\text{square: every exponent even}, \quad \text{cube: every exponent a multiple of } 3
  • Least number to add or subtract for a perfect square

    Nearest squares

    k2≤n<(k+1)2:add (k+1)2−n,subtract n−k2k^2 \le n < (k + 1)^2: \quad \text{add } (k + 1)^2 - n, \quad \text{subtract } n - k^2

Watch out for (5)

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