Question

A number which when divided by 10 leaves a remainder of 9, when divided by 9 leaves a remainder of 8, and when divided by 8 leaves a remainder of 7, is.
(a) 1539
(b) 5139
(c) 2519
(d) 9413

Answer 1

Hint: For solving the above question we will use the method of option elimination. As it is given that when the number is divided by 10 it leaves the remainder 9, i.e., the last digit of the number is 9. So, the option (d) is eliminated. Similarly, use the divisibility rule of 9 which states that the sum of the digits is divisible by 9 and the divisibility rule of 8 according to which a number is divisible by 8 if the last number formed by the last three digits of the number is divisible by 8.

Complete step-by-step answer:
For solving the above question we will use the method of option elimination. The first condition we would use is when the number is divided by 10 it leaves the remainder 9. We know that a number is said to be divisible by 10 if its unit digit is 0. As it is given that 9 is the remainder, the last digit of the number must be 9. So, option (d) cannot be the answer.
The next condition given is that when the number is divided by 9 it leaves remainder 8. So, if we add one to the number then it must be divisible by 9 and for a number to be divisible by 9, the sum of its digit must be divisible by 9 as well. So, let us first check option (a) 1539. If we add one to it, we get 1540 the sum of whose digits is 10, which is not divisible by 9. So, option (a) cannot be the answer.
Next let us check option (b) 5139. If we will add one to it, we get 5140, the sum of whose digit is again 10. So, it is also not the answer.
Now let us check option (c) 2519. If we add one to it, we get 2520. The sum of the digits of 2520 is 9, which is a multiple of 9. So, this can be an answer. Let us further check this option for the third condition to confirm that this is the answer.
So, according to the third condition, the number when divided by 8, leaves the remainder 7. So, if we add 1 to the number, the number must be divisible by 8 and according to the divisibility rule of 8, a number is divisible by 8 if the last number formed by the last three digits of the number is divisible by 8. So, for 2520, the number formed by the last 3 digits is 520 and it is divisible by 8, so 2520 is also divisible by 8.
Hence, we can conclude that option (c) is the answer to the above question.



Note: Whenever it is given that a number a when divided by number b leaves remainder c you can either subtract c from a or add a number such that the remainder becomes equal to b so that the number becomes divisible by b. So, you can take either of the two ways, but be wise and select the way which is easier to apply, as we did in the above question by adding 1.