# Is there a systematic way to determine the number of numbers between 10 and, say, 50, divisible by their units digits?

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I saw a variation on this recently... somewhere. Not allowed to say where. But I ended up writing out some numbers and finding a pattern.

For example...

- anything divisible by 11 is divisible by the units digit, so that includes
**11, 22, 33, and 44.**
- anything that is a multiple of 12 counts, so
**12, 24, 36, and 48,** but of course not 60.
- anything ending in 1 counts, so
**21, 31, and 41**
- anything ending in 2 counts, so
**32 and 42.**
- only 33 ends in 3 and is also divisible by 3, so that doesn't count.
- numbers ending in 4 count if I didn't say them, nothing really.
- anything ending in 5 counts, so
**15, 25, 35, and 45.**
- the only number that counts that ends in 6 is 36, but I already said it.
- nothing ending in 7 counts until 77, which is larger than 50.
- nothing ending in 8 counts since I already said 48.
- nothing ending in 9 counts until 99, which is larger than 50.
- and nothing ending in 0 counts.

And I think that's all of them, so **17** numbers. Which is so abnormal, really.

I saw a variation on this recently... somewhere. Not allowed to say where. But I ended up writing out some numbers and finding a pattern.

For example...

- anything divisible by 11 is divisible by the units digit, so that includes
**11, 22, 33, and 44.** - anything that is a multiple of 12 counts, so
**12, 24, 36, and 48,**but of course not 60. - anything ending in 1 counts, so
**21, 31, and 41** - anything ending in 2 counts, so
**32 and 42.** - only 33 ends in 3 and is also divisible by 3, so that doesn't count.
- numbers ending in 4 count if I didn't say them, nothing really.
- anything ending in 5 counts, so
**15, 25, 35, and 45.** - the only number that counts that ends in 6 is 36, but I already said it.
- nothing ending in 7 counts until 77, which is larger than 50.
- nothing ending in 8 counts since I already said 48.
- nothing ending in 9 counts until 99, which is larger than 50.
- and nothing ending in 0 counts.

And I think that's all of them, so **17** numbers. Which is so abnormal, really.

##### 1 Answer

The number of numbers between

where

#### Explanation:

This is equivalent to asking how many integers

Note that

All that remains, then, is to go through each

This concludes each case, and so, adding them up, we get, as concluded in the question,

In a shorter, easier to calculate notation, using the observations above, we can write the number of integers between

where