Question : M is the largest 4-digit number which, when divided by 4, 5, 6, and 7, leaves the remainder as 2, 3, 4, and 5, respectively. What will be the remainder when M is divided by 9?
Option 1: 2
Option 2: 1
Option 3: 3
Option 4: 6
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Correct Answer: 1
Solution :
The least common multiple (LCM) of 4, 5, 6, and 7 is 420.
The required largest 4-digit number will be a multiple of 420.
Remainder of $\frac{9999}{420}$ is 339.
The largest 4-digit number will be 9999 – 339 = 9660 which is divisible by 4, 5, 6, and 7.
Now, as given, when the number is divided by 4, 5, 6, and 7, it leaves 2, 3, 4, and 5 as remainders.
The difference between each divisor and the remainder is 2, as shown below.
| Divisor | Remainder | Difference |
| 4 | 2 | 2 |
| 5 | 3 | 2 |
| 6 | 4 | 2 |
| 7 | 5 | 2 |
Hence, the required number is 9660 – 2 = 9658
When 9658 is divided by 9, it will leave the remainder 1.
Hence, the correct answer is 1.
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