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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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