A number when divided successively by 4 and 5 leaves remainders 1 and 4 respectively. When it is successively divided by 5 and 4, then the respective. remainders will be

Options

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

Answer

To solve the problem step-by-step, we need to find a number that, when divided successively by 4 and 5, leaves remainders of 1 and 4 respectively. Then, we will determine the remainders when this number is divided successively by 5 and 4.

### Step 1: Set up the equations based on the given conditions.
Let the number be \( x \).

1. When \( x \) is divided by 4, it leaves a remainder of 1:
\[ x \equiv 1 \ (\text{mod} \ 4) \]
This means \( x = 4k + 1 \) for some integer \( k \).

2. When \( x \) is divided by 5, it leaves a remainder of 4:
\[ x \equiv 4 \ (\text{mod} \ 5) \]
This means \( x = 5m + 4 \) for some integer \( m \).

### Step 2: Substitute the first equation into the second equation.
From the first equation, we can substitute \( x \) in the second equation:
\[ 4k + 1 \equiv 4 \ (\text{mod} \ 5) \]
Subtracting 1 from both sides gives:
\[ 4k \equiv 3 \ (\text{mod} \ 5) \]

### Step 3: Solve for \( k \).
To solve \( 4k \equiv 3 \ (\text{mod} \ 5) \), we can find the multiplicative inverse of 4 modulo 5. The inverse of 4 is 4 itself since \( 4 \times 4 = 16 \equiv 1 \ (\text{mod} \ 5) \).

Multiplying both sides of the equation by 4:
\[ k \equiv 4 \times 3 \ (\text{mod} \ 5) \implies k \equiv 12 \ (\text{mod} \ 5) \implies k \equiv 2 \ (\text{mod} \ 5) \]
This means \( k = 5n + 2 \) for some integer \( n \).

### Step 4: Substitute \( k \) back to find \( x \).
Substituting \( k \) back into the equation for \( x \):
\[ x = 4(5n + 2) + 1 = 20n + 8 + 1 = 20n + 9 \]
Thus, \( x \equiv 9 \ (\text{mod} \ 20) \).

### Step 5: Find the smallest positive value of \( x \).
The smallest positive value for \( n = 0 \) gives:
\[ x = 9 \]

### Step 6: Check the conditions.
- \( 9 \div 4 = 2 \) remainder \( 1 \) (correct)
- \( 9 \div 5 = 1 \) remainder \( 4 \) (correct)

### Step 7: Now find the remainders when \( x = 9 \) is divided successively by 5 and 4.
1. First, divide by 5:
\[ 9 \div 5 = 1 \quad \text{remainder} \ 4 \]

2. Next, take the result and divide by 4:
\[ 4 \div 4 = 1 \quad \text{remainder} \ 0 \]

### Final Result:
When the number 9 is divided successively by 5 and 4, the remainders are 4 and 0 respectively.

### Summary:
- The number is \( 9 \).
- The remainders when divided by 5 and 4 are \( 4 \) and \( 0 \).