Uses of Prime Factor Decomposition (Edexcel IGCSE Maths)

Spec point — 1.4D

Uses of prime factor decomposition

How can I use PFD to identify a square or cube number?

• If all the indices in the prime factor decomposition of a number are even, then that number is a square number

  • E.g. The prime factor decomposition of 7056 is 2⁴ × 3² × 7²

  • All powers are even so it must be a square number

    • It can be written as (2² × 3 × 7)²

• If all the indices in the prime factor decomposition of a number are multiples of 3, then that number is a cube number

  • E.g. The prime factor decomposition of 1728000 is 2⁹ × 3³ × 5³

  • All powers are multiples of 3 so it must be a cube number

    • It can be written as (2³ × 3 × 5)³

How can I use PFD to find the square root of a square number?

• Write the number in its prime factor decomposition

  • All the indices should be even if it is a square number

• For example, to find the square root of 144 = 2⁴ × 3²

  • Halve all of the indices

    • 2² × 3
    • So $\sqrt{2^{4}\times3^{2}}=2^{2}\times3$

• This is the prime factor decomposition of the square root of the number

  • To find it as an integer, multiply the prime factors together
  • 2² × 3 = 12, so the square root of 144 is 12

How can I use PFD to find the exact square root of a number?

• If the number is not a square number, its exact square root can still be found using its prime factor decomposition

• Write the number in its prime factor decomposition

  • $1440=2^{5}\times3^{2}\times5$

• Rewrite the prime factor decomposition with as many even indices as you can

  • E.g. 2³ = 2² × 2, or 5⁷ = 5⁶ × 5
  • $1440=2^{4}\times2\times3^{2}\times5$

• Collect the terms with even powers together

  • $1440=2^{4}\times3^{2}\times2\times5$

• Square root both sides

  • $\sqrt{1440}=\sqrt{2^{4}\times3^{2}\times2\times5}$

• Using the rule $\sqrt{ab}=\sqrt{a}\sqrt{b}$, apply the square root to the terms with the even indices separately to the terms with odd indices

  • $\sqrt{1440}=\sqrt{2^{4}\times3^{2}}\times\sqrt{2\times5}$

• Simplify to find your answer, remembering that $\sqrt{a^{2}b^{2}}=ab$

  • $\sqrt{1440}=2^{2}\times3\times\sqrt{10}$
  • $\sqrt{1440}=12\sqrt{10}$
  • $12\sqrt{10}$ is the exact square root of $1440$

Worked Example $N=2^{3}\times3^{2}\times5^{7}$ and $AN=B$ where $A$is an integer and $B$ is a non-zero square number.

Find the smallest value of $A$.

Answer:

Substitute N = 2³ × 3² × 5⁷* into the formula AN = *B

A(2**³ × 3² × 5⁷**) = B

To be a square number, the prime factors of AN must all have even powers

Consider the prime factors A needs to have to make all the values on the left hand side have even powers

(2 × 5) (2*³ × 3² × 5⁷) = B **

2*⁴ × 3² × 5⁸ **= B

So A, when written as a product of its prime factors, is 2 × 5

Make sure you write A as an integer value in the answer

Final answer: A = 10**