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Which one of the following numbers is completely divisible by \[45\] ?
A. \[1,81,560\]
B. \[3,31,145\]
C. \[2,02,860\]
D. \[20,38,550\]

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Last updated date: 19th Jul 2024
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Answer
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Hint: In order to solve this question, we will go through by applying the divisibility rule of \[45\] to check which number is completely divisible by \[45\] . And we know that if the number is divisible by \[5\] and \[9\] then the number is divisible by \[45\] . So here we will use the divisibility test of \[5\] and \[9\] and check each option. Then we will conclude which option satisfies the divisibility test of both \[5\] and \[9\] . And hence we will get our required result.

Complete step by step answer:
Here in the question, we have to find the number which is divisible by \[45\]. So, first of all let’s understand the divisible rule of \[45\] i.e., divisibility rule of \[45\] is the number divisible by both \[5\] and \[9\]. Now we know that the divisibility rule of \[5\] is if the number ends with \[0\] or \[5\] then it is divisible by \[5\]. And the divisibility rule of \[9\] is if we add up all the digits and the addition is divisible by \[9\] then the number itself is also divisible by \[9\]

Now by using the above definitions we will check each option.
Option (a) is \[1,81,560\]
First let’s check the divisibility rule of \[5\].
Here, the last digit is \[0\] which means it is divisible by \[5\].
Now let’s check the divisibility rule of \[9\].
The sum of the digits is \[1 + 8 + 1 + 5 + 6 + 0 = 21\] and we know \[21\] is not divisible by \[9\].
Since, the number is only divisible by \[5\] and not \[9\].
Hence, the given number is not divisible by \[45\].

Now option (b) is \[3,31,145\]
So first let’s check the divisibility rule of \[5\].
Here, the last digit is \[5\] which means it is divisible by \[5\].
Now let’s check the divisibility rule of \[9\]. The sum of the digits is \[3 + 3 + 1 + 1 + 4 + 5 = 17\] and we know \[17\] is not divisible by \[9\].
Since, the number is only divisible by \[5\] and not \[9\].
Hence, the given number is not divisible by \[45\].

Now option (c) is \[2,02,860\]
So first let’s check the divisibility rule of \[5\].
Here, the last digit is \[0\] which means it is divisible by \[5\].
Now let’s check the divisibility rule of \[9\].
The sum of the digits is \[2 + 0 + 2 + 8 + 6 + 0 = 18\] and we know \[18\] is divisible by \[9\].
Since, the number is divisible by both \[5\] and \[9\].
Hence, the given number is divisible by \[45\].

Now let’s check the option (d) as well
Given number is \[20,38,550\].
So first let’s check the divisibility rule of \[5\].
Here, the last digit is \[0\] which means it is divisible by \[5\].
Now let’s check the divisibility rule of \[9\].
The sum of the digits is \[2 + 0 + 3 + 8 + 5 + 5 + 0 = 23\] and we know \[23\] is not divisible by \[9\].
Since, the number is divisible by only \[5\] and not \[9\].
Hence, the given number is not divisible by \[45\].
Therefore, from the above calculation we conclude that the number \[2,02,860\] is divisible by \[45\].

Hence, option C is the correct answer.

Note: Whenever we face such types of problems, the key concept for solving the question is to go through the divisibility rule of that number. Also remember that co-primes of any number can be used to check the divisibility of very large numbers. Also, the easiest way to verify the result is direct division of dividend through divisor. And if the remainder comes out to be zero, then it can be said that the dividend is completely divisible by the divisor.