Find the largest 3 digit number given remainder 1 and 5, when divided by 5 & 8

Options

A. 996
B. 991
C. 981
D. 986

Answer

To solve the problem of finding the largest 3-digit number that gives a remainder of 1 when divided by 5 and a remainder of 5 when divided by 8, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the largest 3-digit number**: The largest 3-digit number is 999.

2. **Check the conditions**: We need to find a number \( N \) such that:
- \( N \mod 5 = 1 \)
- \( N \mod 8 = 5 \)

3. **Start from the largest 3-digit number**: We will check the largest 3-digit numbers starting from 999 and moving downwards.

4. **Check 999**:
- **Dividing by 5**:
\[ 999 \div 5 = 199 \quad \text{(Remainder: 4)} \]
Since the remainder is 4, it does not satisfy the condition \( N \mod 5 = 1 \).

5. **Check 998**:
- **Dividing by 5**:
\[ 998 \div 5 = 199 \quad \text{(Remainder: 3)} \]
Since the remainder is 3, it does not satisfy the condition.

6. **Check 997**:
- **Dividing by 5**:
\[ 997 \div 5 = 199 \quad \text{(Remainder: 2)} \]
Since the remainder is 2, it does not satisfy the condition.

7. **Check 996**:
- **Dividing by 5**:
\[ 996 \div 5 = 199 \quad \text{(Remainder: 1)} \]
This satisfies the condition \( N \mod 5 = 1 \).
- **Dividing by 8**:
\[ 996 \div 8 = 124 \quad \text{(Remainder: 4)} \]
Since the remainder is 4, it does not satisfy the condition \( N \mod 8 = 5 \).

8. **Check 995**:
- **Dividing by 5**:
\[ 995 \div 5 = 199 \quad \text{(Remainder: 0)} \]
This does not satisfy the condition.

9. **Check 994**:
- **Dividing by 5**:
\[ 994 \div 5 = 198 \quad \text{(Remainder: 4)} \]
This does not satisfy the condition.

10. **Check 993**:
- **Dividing by 5**:
\[ 993 \div 5 = 198 \quad \text{(Remainder: 3)} \]
This does not satisfy the condition.

11. **Check 992**:
- **Dividing by 5**:
\[ 992 \div 5 = 198 \quad \text{(Remainder: 2)} \]
This does not satisfy the condition.

12. **Check 991**:
- **Dividing by 5**:
\[ 991 \div 5 = 198 \quad \text{(Remainder: 1)} \]
This satisfies the condition \( N \mod 5 = 1 \).
- **Dividing by 8**:
\[ 991 \div 8 = 123 \quad \text{(Remainder: 7)} \]
This does not satisfy the condition.

13. **Check 990**:
- **Dividing by 5**:
\[ 990 \div 5 = 198 \quad \text{(Remainder: 0)} \]
This does not satisfy the condition.

14. **Check 989**:
- **Dividing by 5**:
\[ 989 \div 5 = 197 \quad \text{(Remainder: 4)} \]
This does not satisfy the condition.

15. **Check 988**:
- **Dividing by 5**:
\[ 988 \div 5 = 197 \quad \text{(Remainder: 3)} \]
This does not satisfy the condition.

16. **Check 987**:
- **Dividing by 5**:
\[ 987 \div 5 = 197 \quad \text{(Remainder: 2)} \]
This does not satisfy the condition.

17. **Check 986**:
- **Dividing by 5**:
\[ 986 \div 5 = 197 \quad \text{(Remainder: 1)} \]
This satisfies the condition \( N \mod 5 = 1 \).
- **Dividing by 8**:
\[ 986 \div 8 = 123 \quad \text{(Remainder: 2)} \]
This does not satisfy the condition.

18. **Check 985**:
- **Dividing by 5**:
\[ 985 \div 5 = 197 \quad \text{(Remainder: 0)} \]
This does not satisfy the condition.

19. **Check 984**:
- **Dividing by 5**:
\[ 984 \div 5 = 196 \quad \text{(Remainder: 4)} \]
This does not satisfy the condition.

20. **Check 983**:
- **Dividing by 5**:
\[ 983 \div 5 = 196 \quad \text{(Remainder: 3)} \]
This does not satisfy the condition.

21. **Check 982**:
- **Dividing by 5**:
\[ 982 \div 5 = 196 \quad \text{(Remainder: 2)} \]
This does not satisfy the condition.

22. **Check 981**:
- **Dividing by 5**:
\[ 981 \div 5 = 196 \quad \text{(Remainder: 1)} \]
This satisfies the condition \( N \mod 5 = 1 \).
- **Dividing by 8**:
\[ 981 \div 8 = 122 \quad \text{(Remainder: 5)} \]
This satisfies the condition \( N \mod 8 = 5 \).

23. **Conclusion**: The largest 3-digit number that satisfies both conditions is **981**.