When a number is successively divided by 4 and 5. The remainder obtained are 1 and 4 respectively. when the same number is successively divided by 5 and 4, the remainder obtained are -?

Options

A. 2 , 3
B. 3, 4
C. 2, 1
D. 3, 2

Answer

To find the remainders when the same number is successively divided by 5 and 4, we can follow these steps:

### Step 1: Understand the problem
We know that when a number \( N \) is divided by 4, the remainder is 1. This can be expressed as:
\[ N \equiv 1 \ (\text{mod} \ 4) \]

When the same number \( N \) is divided by 5, the remainder is 4. This can be expressed as:
\[ N \equiv 4 \ (\text{mod} \ 5) \]

### Step 2: Set up the equations
From the above congruences, we can express \( N \) in terms of a variable:
1. From \( N \equiv 1 \ (\text{mod} \ 4) \), we can write:
\[ N = 4k + 1 \] for some integer \( k \).

2. From \( N \equiv 4 \ (\text{mod} \ 5) \), we can write:
\[ N = 5m + 4 \] for some integer \( m \).

### Step 3: Equate the two expressions for \( N \)
Now we have two expressions for \( N \):
\[ 4k + 1 = 5m + 4 \]

### Step 4: Rearrange the equation
Rearranging gives:
\[ 4k - 5m = 3 \]

### Step 5: Solve for integer values
We need to find integer solutions for \( k \) and \( m \). We can try different values for \( m \) to find corresponding \( k \):
- If \( m = 1 \):
\[ 4k - 5(1) = 3 \]
\[ 4k = 8 \]
\[ k = 2 \]

So one solution is \( k = 2 \) and \( m = 1 \).

### Step 6: Find \( N \)
Using \( k = 2 \):
\[ N = 4(2) + 1 = 8 + 1 = 9 \]

### Step 7: Verify the solution
- Check \( N \) when divided by 4:
\[ 9 \div 4 = 2 \quad \text{remainder} = 1 \] (correct)

- Check \( N \) when divided by 5:
\[ 9 \div 5 = 1 \quad \text{remainder} = 4 \] (correct)

### Step 8: Find the remainders when divided by 5 and 4
Now we need to find the remainders when \( N = 9 \) is divided by 5 and then by 4:
1. When \( N = 9 \) is divided by 5:
\[ 9 \div 5 = 1 \quad \text{remainder} = 4 \]

2. When \( N = 9 \) is divided by 4:
\[ 9 \div 4 = 2 \quad \text{remainder} = 1 \]

### Final Answer
Thus, the remainders obtained when the same number is successively divided by 5 and 4 are:
- Remainder when divided by 5: 4
- Remainder when divided by 4: 1