Anholm
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Hello, could somebody please explain how I could go about solving this question (without using a calculator)

Find the smallest number that 1500 needs to be multiplied by in order to make a square number.

I'm given the options
45,
10,
6,
30,
15
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bl0bf1sh
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are any of the options very similar to 1500? perhaps start the same? possibly a different number of 0s on the end? (e.g. 500x5 = 50x50)
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old_engineer
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(Original post by Anholm)
Hello, could somebody please explain how I could go about solving this question (without using a calculator)

Find the smallest number that 1500 needs to be multiplied by in order to make a square number.

I'm given the options
45,
10,
6,
30,
15
I would start by finding the prime factors of 1,500, which you can do by dividing by 2 as many ties as possible, then by 3, then by 5 etc. Then have a think about what the indices of the prime factors would need to look like if the number is to have an integer square root.
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Anholm
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(Original post by old_engineer)
I would start by finding the prime factors of 1,500, which you can do by dividing by 2 as many ties as possible, then by 3, then by 5 etc. Then have a think about what the indices of the prime factors would need to look like if the number is to have an integer square root.
Right, so that would make 5^3 × 3 × 2^2
That is fine but I've never been taught where to go from there...
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the bear
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express 1500 as ? x ? x ? x ? x ? x ?

where the question marks are prime numbers.

that will make it easy.
Last edited by the bear; 1 month ago
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old_engineer
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(Original post by Anholm)
Right, so that would make 5^3 × 3 × 2^2
That is fine but I've never been taught where to go from there...
You could observe that 2^2 has an integer square root but 5^3 doesn't. But then 5^4 might have an integer square root......
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Anholm
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(Original post by the bear)
express 1500 as ? x ? x ? x ? x ? x ?

where the question marks are prime numbers.

that will make it easy.
Ok,
5×5×5×3×2×2
I guess I then single out the pairs leaving 5×3=15
Is that how it's done or have I just made something up?
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the bear
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(Original post by Anholm)
Ok,
5×5×5×3×2×2
I guess I then single out the pairs leaving 5×3=15
Is that how it's done or have I just made something up?
your method is fine... a square number will have its prime factors in two equal groups...

here we have two groups of 5 x 2, but the remaining factors ( 5 x 3 ) are not yet in equal groups, so we must introduce an extra 5 and an extra 3
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RogerOxon
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(Original post by Anholm)
Hello, could somebody please explain how I could go about solving this question (without using a calculator)

Find the smallest number that 1500 needs to be multiplied by in order to make a square number.

I'm given the options
45,
10,
6,
30,
15
1500 = 2^2 * 3 * 5^3
To get a perfect square, you need an even number of each prime factor. Which prime factors do you need to multiply by to get that?
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RogerOxon
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(Original post by Anholm)
Ok,
5×5×5×3×2×2
I guess I then single out the pairs leaving 5×3=15
Is that how it's done or have I just made something up?
That's correct.

For bonus marks, calculate the square root of the result, without a calculator.
Last edited by RogerOxon; 1 month ago
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