# Quantitative: The Basics II: Factorisation, Square numbers and Powers

by on Thursday, 6 January, 2011

Here’s another question:

What is the smallest positive integer y such that 71,400 multiplied by y is the square of a positive integer?

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We could spend hours doing this by trial and error. But let’s consider what we know about square numbers, and about their factors, specifically their prime factors.

Example 1
36 = 6 x 6. Break it down into the product of its prime factors:

36 = 2 x 3 x 2 x 3, or 2² x 3²

Example 2
576 = 24 x 24

24 = 2 x 2 x 2 x 3 so 24² = 2 x 2 x 2 x 3 x 2 x 2 x 2 x 3 or 2^6 x 3²

N.B. 2^6 means 2 to the power of 6. A power is sometimes called an index or an exponent. All refer to the little number above and to the right of an integer.

What you will notice is that when you break a square number down into the product of its prime factors, each of those prime factors will be raised to an even power. This makes sense: even numbers are divisible by 2, and each of those powers will have to be divided by 2 to find the square root.

Example
The square root of 3² x 7^18 x 13^6 is 3 x 7^9 x 13^3.

And remember that when you multiply, you add powers:

Example
13^3 x 13^3 = 13^(3+3) = 13^6

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How does this help with our question? Start by factorising 71,400:

71,400 = 100 x 714 = 2 x 5 x 2 x 5 x 714 = 2² x 5² x 2 x 357 = 2^3 x 5² x 7 x 51 = 2^3 x 5² x 7 x 3 x 17

Now, 5² contains an even power. But the other prime factors don’t. So to get a square number, we will need to multiply once by 2, and then by 7, 3 and 17.

Answer: 2 x 7 x 3 x 17 = 714.

N.B. There are different ways of factorising. Taking out powers of 10 (and then 5 if you like) is always a good way to start: just remember to break down the 10 into 2 x 5. You can also work up through the prime numbers: 2, 3, 5, 7, 11 is normally enough.

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There’s a trick to see whether a number is divisible by 3. If the sum of its digits is a multiple of 3 then the number is also a multiple of 3. This sounds more complicated than it is. Take 357:

3 + 5 + 7 = 15

15 is a multiple of 3

so 357 is a multiple of 3

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Practice
Write out and become familiar with square numbers from 2² = 4 up to 25² = 625

Do the same with the first few cube numbers

What number is both a square and a cube?

Change 71,400 in the question above to a) 2,835   b) 377,300   c) 168,168;   what is y in each case?

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