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If the number 24753x is completely divisible by 72 then which of the following numbers should replace x
(a) 4
(b) 5
(c) 6
(d) 7

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Last updated date: 13th Jun 2024
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Answer
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Hint: Divisibility rule of 8: - Last three-digit of number should be divisible by 8 then the whole number is divisible by 8.
Divisibility rule of 9: -
The Sum of all digits of a number should be multiple of 9 or divisible by 9.

Complete step-by-step solution:
We will first of all factor 72.
Since 72 is a composite number it has more than 2 factors.
8 and 9 are coming under factors of 72.
So for any number to be divisible by 72, it should be divisible by 9 & 8.
Now we will study the divisible rules of 8 and 9.
Divisibility rule of 8: -
The last three digits of the number should be divisible by 8 then the whole number is divisible by 8.
Divisibility rule of 9: -
The Sum of all digits of a number should be multiple of 9 or divisible by 9.
Now we have the number as 24753x.
Sum is 2 + 4 + 7 + 5 + 3 + x, for it to be divisible by 9 we must have, sum = multiple of 9.
\[\Rightarrow \] 2 + 4 + 7 + 5 + 3 + x = multiple of 9.
\[\Rightarrow \] 21 + x = multiple of 9.
Now we will consider all options separately.
Option (a), Let x = 4,
\[\Rightarrow \] 21 + 4 = 25, not a multiple of 9.
So, option (a) is correct.
Option (b), Let x = 5.
\[\Rightarrow \] 21 + 5 = 26, not a multiple of 9.
So, option (b) is incorrect.
Option (c), Let x = 6.
\[\Rightarrow \] 21 + 6 = 27, yes a multiple of 9.
 So, option (c) is correct.
Option (d), Let x = 7, then,
\[\Rightarrow \] 21 + 7 = 28, not a multiple of 9.
Wrong option.
Also when x = 6, then the last three digits of 24753x are 536 which is divisible by 8.
So, x = 6 is correct answer which is option (c).

Note: Another way to solve this question can be checked for divisibility of 8 on all options. That can be long but will anyway give the same result.
Option (a), let x = 4
Then 247534 has last three-digit as 534.
Clearly, \[\dfrac{534}{8}=66.75\].
So, 534 is not divisible by 8.
Hence option (a) is wrong.
Option (b), let x = 5
Then 247535 has the last three digits as 535.
Clearly, \[\dfrac{535}{8}=66.87\].
So, 535 is not divisible by 8.
Option (b) is wrong.
Option (c), let x = 6
Then 247536 has the last three digits as 536.
Clearly, \[\dfrac{536}{8}=67\].
So, 536 is divisible by 8.
Hence option (c) is correct.
Option (d), let x = 7.
Then 537 is not divisible by 8.
Option (d) wrong.