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From given data

When the number is divided by 18, it leaves a remainder of 7.

When the number is divided by 21, it leaves a remainder of 10.

When the number is divided by 24, it leaves a remainder of 13.

Observing the constant difference of 11 between divisor and remainder:

We can conclude that if 11 is subtracted from the LCM of 18, 21 and 24, then it will leave a remainder of 7 when divided by 18, a remainder of 10 when divided by 21 and a remainder of 13 when divided by 24.

When 11 is subtracted from any multiple of LCM, it will give the same result.

Now,

18=2\times 3\times 3

21=3\times 7

24=2\times 2\times 2\times 3

\therefore LCM\ of\ 18,\ 21\ and\ 24 = 504

So, any number of the form (504n – 11) will give the required remainders, when n is an integer.

But, this number also has to be divisible by 23.

Using trial and error method, for n = 6, we find that-

504n-11=504\times 6-11=3024-11=3013

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