Video: GCSE Mathematics Foundation Tier Pack 3 • Paper 1 • Question 11

GCSE Mathematics Foundation Tier Pack 3 • Paper 1 • Question 11

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Video Transcript

Find four unique factors of 64 that sum to 46.

The factors of a number are the numbers that multiply together to give that number. So we’ll begin by listing out the factors of 64. First of all, 64 can be written as one multiplied by 64. 64 is also an even number and dividing 64 by two gives 32. So we can also write 64 as two multiplied by 32. Now, 64 can’t be divided by three. As we can see that if we use a short division method, then there’s a remainder of one. So three isn’t a factor of 64.

However, four is. If we divide 64 by four, we get 16. So we can write 64 as four multiplied by 16. 64 can’t be divided by five, six, or seven which we could again see using a short division method. But it is equal to eight multiplied by eight. Which you probably recognize as 64 is a square number, eight squared.

Now, as these two factors, eight and eight, are the same size, this means that we found all of the factors of 64. If there are any factors larger than eight, they would have to be paired with numbers smaller than eight in order to multiply together to give 64. As we’ve already checked all of the numbers smaller than eight, this is our complete list of factor pairs. So the full list of factors of 64 is one, two, four, eight, 16, 32, and 64.

We’ve been asked to find four unique factors, so that means different factors, of 64 that sum to 46. Immediately, we can eliminate 64 itself as this is bigger than 46. We can also eliminate one as this is the only odd factor of 64. And if we add an odd number to an even number, we’d always end up with an odd result. So as all the other factors are even, one can’t be one of the factors that we’re looking for.

We’ve therefore narrowed our list down to just five factors. Now, as 46 is quite a large number relative to all of these factors, we need to start off with quite a big number. So we’ll assume that 32 is one of the factors we’re looking for. The difference between 46 and 32 is 14, which we’re now looking to be the sum of three unique factors.

16 can’t be one of these factors as this is bigger than 14. So we’re left with two, four, and eight. Two plus four plus eight is indeed equal to 14. That’s because two plus four is six and six plus eight is 14. So these are indeed the three factors that we’re looking for. So the four unique factors of 64 that sum to 46 are two, four, eight, and 32.

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