The Lowest Common Multiple of x and 180 is 360. Find the smallest possible value of x.
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So you want a number which is a factor of 360 but not of 180.
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In other words a number you can divide 360 by without leaving a remainder, but dividing 180 with it will leave a remainder.
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Start with the smallest factor of 360 and check if you can also divide 180 by it:
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2, 3, 4, 5, 6 are factors of both 360 and 180.
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But the next factor of 360 which is 8 is not a factor of 180.
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Therefore the answer is 8.
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If you continue in the same way:
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9, 10, 12, 15, 18, 20 are factors of both 360 and 180.
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But the next factor of 360 which is 24 is not a factor of 180.
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So if x was 24, the LCM would also be 360, but of course you want the lowest value of x, which is 8.
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The highest possible value of x would be 360.
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If x was 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 45, 60, 90, or 180, the LCM would be 180 and not 360.
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Sometimes an easy way of finding the LCM of 2 numbers can be to do the times table of the bigger number until you get a number which can be divided by the smaller number.
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For example if you want to find the LCM of 12 and 15, consider the 15 times table: 15, 30, 45, 60.
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15, 30, 45 cannot be divided by 12 but 60 can, so the LCM is 60.
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Of course, the proper mathematical way is to first break down both 12 and 15 as products of their prime factors:
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12 = 2 x 2 x 3
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15 = 3 x 5
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LCM = 2 x 2 x 3 x 5 = 60.
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You only include 3 once because it is common for both 12 and 15.
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The Highest Common Factor (HCF), the biggest number you can divide both 12 and 15 by would be 3 because that is the only common factor.
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To find the HCF of 2 numbers, you multiply all the prime factors which are common.
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This might be easier to read!