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Making squares and cubes: smallest n problems

Smallest Integer N Problems Core + Extended 6 min read

On this page
  1. In this note
  2. Formula bank
  3. Squares and cubes in prime factors
  4. Making a square or a cube
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 1.1: squares, cubes and prime factors
  • 1.1Find square and cube roots using prime factors.
  • 1.1Find the smallest number to multiply or divide by to make a square or a cube.

Before you start: prime factors.

Formula bank

Perfect squareLearn itevery index in the prime factorisation is EVEN\(2^{4}\times3^{2}\times7^{2}=84^{2}\).Use it when "is a square number".
Perfect cubeLearn itevery index is a MULTIPLE OF 3\(3^{3}\times5^{3}=15^{3}\).Use it when "is a cube number".
Square / cube rootLearn ithalve every index / divide every index by 3Use it when a root without a calculator.
Smallest multiplierLearn itthe fewest extra primes to fix every indexUse it when "smallest n such that 180n is a square".

Squares and cubes in prime factors

Squaring doubles every index: \(84^{2}=(2^{2}\times3\times7)^{2}=2^{4}\times3^{2}\times7^{2}\). So a number is a perfect square exactly when every index is even, and a perfect cube when every index is a multiple of 3.

General methodHow to find a square or cube root using prime factors

  1. Write the number in prime factors, index form.
  2. Square root: halve every index. Cube root: divide every index by 3.
  3. Multiply out.

Example \(1764=2^{2}\times3^{2}\times7^{2}\). Find \(\sqrt{1764}\).

Halve each index.

  1. \(2\times3\times7\)

Answer 42

Try it yourself

1
\(\sqrt{2^{6}\times3^{2}}\)
Show answer

Answer 24

2
Use \(7056=2^{4}\times3^{2}\times7^{2}\) to find \(\sqrt{7056}\).
Show answer

Answer 84

3
\(\sqrt[3]{3^{3}\times5^{3}}\)
Show answer

Answer 15

Making a square or a cube

General methodHow to find the smallest multiplier (or divisor)

  1. Write the number in prime factors, index form.
  2. Check each index: even (for a square)? a multiple of 3 (for a cube)?
  3. Multiplier: add the FEWEST of each prime to reach the next even number / multiple of 3. Divisor: take out the fewest to go down.
  4. Multiply those primes together. Check your answer.

Example Find the smallest whole number n such that 180n is a cube number.

\(180=2^{2}\times3^{2}\times5\): bring every index up to 3.

  1. need one 2, one 3 and two 5s
  2. \(n=2\times3\times5^{2}\)
  3. \(180\times150=27\,000=30^{3}\)

Answer 150

Try it yourself

1
Smallest n so that 180n is a square?
Show answer

Answer 5

2
Smallest n so that 48n is a square?
Hint
\(48=2^{4}\times3\).
Show answer

Answer 3

3
Smallest k so that 252 ÷ k is a square?
Hint
\(252=2^{2}\times3^{2}\times7\).
Show answer

Answer 7

Ways they test it

Type 1 Root from given factors

"\(1764=2^{2}\times3^{2}\times7^{2}\). Use this to find \(\sqrt{1764}\)." (2 marks)

Spot it "use this" with a factorisation.

  1. Halve the indices.
  2. Multiply out.

Type 2 Smallest multiplier

"Find the smallest positive integer n such that 540n is a square number." (2 marks)

Spot it "smallest" + square or cube.

  1. Prime factors (M1).
  2. Product of the missing primes (A1).

For example 540

  1. \(2^{2}\times3^{3}\times5\)
  2. odd: 3³ and 5¹

Answer 15

Type 3 In index form

"\(M=2^{3}\times3^{2}\times5\times7\). RM is a cube. Find the smallest R."

Spot it a number already in index form.

  1. Raise each index to the next multiple of 3.

For example \(2^{3}\times3^{2}\times5\times7\)

  1. need \(3\times5^{2}\times7^{2}\)

Answer 3675

Type 4 Divide to make one

"Find the smallest k such that \(\dfrac{3888}{k}\) is a cube."

Spot it a division.

  1. Take out the extra primes.

For example \(3888=2^{4}\times3^{5}\)

  1. remove one 2 and two 3s

Answer 18

Mistakes to avoid

  • ✗ Fixing only one odd index✓ fix every odd indexAll indices must be even.
  • ✗ n = the number itself✓ only the missing primesSmallest, not any.
  • ✗ Cube: making the indices even✓ multiples of 3That is for squares.

Summary

  • Square: every index even. Cube: every index a multiple of 3.
  • Roots: halve (square) or divide by 3 (cube) every index.
  • Smallest multiplier / divisor: only the primes needed to fix each index.

If you see …

  • "smallest n … square"make every index even.
  • "smallest n … cube"make every index a multiple of 3.

Practice

Practice: easy, medium, hard

Easy

1
Smallest n so 50n is a square?
Show working
  1. \(50=2\times5^{2}\)

Answer 2

2
Smallest n so 24n is a cube?
Show working
  1. \(24=2^{3}\times3\): need \(3^{2}\)

Answer 9

3
Smallest n so 48n is a cube?
Show working
  1. \(2^{4}\times3\): need \(2^{2}\times3^{2}\)

Answer 36

Medium

1
Smallest positive integer n such that 540n is a square?
Guide me

What is it asking? n

What do we know? \(540=2^{2}\times3^{3}\times5\)

Plan fix the odd indices

Show working
  1. one more 3 and one more 5

Answer 15

2
\(N=2^{5}\times3^{2}\times11\). Smallest P so PN is a square?
Guide me

What is it asking? P

What do we know? indices 5, 2, 1

Plan fix the odd ones

Show working
  1. \(2\times11\)

Answer 22

3
Write 2025 as a product of prime factors, then find √2025.
Guide me

What is it asking? the root

What do we know? 2025

Plan \(3^{4}\times5^{2}\), halve

Show working
  1. \(3^{2}\times5\)

Answer 45

Hard

1
\(M=2^{3}\times3^{2}\times5\times7\). R is the smallest integer with RM a cube. Find R.
Guide me

What is it asking? R

What do we know? indices 3, 2, 1, 1

Plan up to 3 each

Show working
  1. \(3\times5^{2}\times7^{2}\)

Answer 3675

PartAnswer / workingMarksGuidance
\(3\times5^{2}\times7^{2}\)M1
3675A1
Total: 2 marks
2
Smallest k such that \(\dfrac{3888}{k}\) is a cube (\(3888=2^{4}\times3^{5}\)).
Guide me

What is it asking? k

What do we know? indices 4 and 5

Plan take out down to 3

Show working
  1. \(2\times3^{2}\)

Answer 18

PartAnswer / workingMarksGuidance
one 2 and two 3sM1
18A1
Total: 2 marks
3
n is the smallest positive integer with 12n a square AND 18n a cube. Find n.
Hint
Let \(n=2^{a}3^{b}\): \(12n=2^{2+a}3^{1+b}\), \(18n=2^{1+a}3^{2+b}\).
Guide me

What is it asking? n

What do we know? two conditions

Plan find a and b that satisfy both

Show working
  1. a = 2 (4 even, 3 a multiple of 3)
  2. b = 1 (2 even, 3 a multiple of 3)
  3. \(2^{2}\times3\)

Answer 12

PartAnswer / workingMarksGuidance
a = 2M1
b = 1M1
12A1
Total: 3 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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