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Find all the common multiples of 10 and nine up to and including 450.

We need to find all the numbers that are multiples of both 10 and nine. Because they’re multiples of two numbers, we call them common multiples of 10 and nine. In other words, they must be in the 10 times table. And they must also be in the nine times table. And we’re told where we need to stop, which is 450. To begin with, let’s think about multiples of nine. What are the multiples of nine up to 10 times nine. We know that adding nine each time is the same as adding 10 and taking away one. And we can use this to spot a pattern with the nine times table. If we add one in the tens column each time and then take away one from the digit in the ones column, we can work out the next multiple of nine each time. add one 10 take away one one, add one 10 take away one one, and so on.

Five nines of 45, six nines of 54, 63, 72, 81, and then 10 times nine, we know, is 90. We can see that the number of tens increases by one each time. And we could also see that their digits in the ones place decrease by one each time. Just a quick way of working out the multiples in line. Multiples of 10 are a lot easier to spot. We know that they always end in a zero. And so the only multiple of nine between nine and 90, that’s also a most worth 10, is 90 itself. This is the same as one lot of 10 times nine. But 90 isn’t actually the first common multiple of 10 and nine. The first multiple that both numbers have in common is actually zero. And we can find that out by multiplying zero times 10 times nine. Zero is in the nine times table. It’s the same as zero times nine. And it’s also in the 10 times table. It’s the same as zero times 10.

So we can see a pattern here to help us find all the common multiples of 10 and nine up to and including 450. We just need to count in 90s. Two lots of 90 or two lots of 10 times nine equals 180. Can you see a pattern with these numbers? They look familiar, don’t they? They’re actually multiples of nine with a zero on the end. In other words, they’re multiples of nine that have been multiplied then by 10. 180 is in the nine times table, it’s the same as 20 times nine, and is also in the 10 times table, it’s the same as 18 times 10. So these numbers are all common multiples of 10 and nine. Let’s continue the pattern. We know three multiplied by nine equals 27. So three more supplied by 10 times nine or 90 equals 270. This is a number that appears in both times tables. Four nines are 36. So four times 90 or four times 10 times nine equals 360.

And we also need to include one more common multiple because we’re told to find all the common multiples of 10 and nine up to and including 450. So we need to include 452 which is the same as five times 90 or five times 10 times nine. We found that all the numbers that are both multiples of 10 and nine are actually multiples of 90. So we could just count in 90s. We knew we had to include zero because that’s a number in both the 10 and the nine times tables. We knew we also had to include 450. So all the common multiples of 10 and nine up to and including 450 are zero, 90, 180, 270, 360, and 450.