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?