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

We are given a number that, when divided successively by 4 and 5, leaves remainders 1 and 4, respectively. We need to find the remainders when this number is divided successively by 5 and 4.

Step 1: Express the conditions algebraically
Let the number be denoted as N.

When N is divided by 4, the remainder is 1. This means: N = 4k + 1 for some integer k.

When N is divided by 5, the remainder is 4. This means: N = 5m + 4 for some integer m.

Step 2: Use the system of congruences
We have two congruences:

N ≡ 1 (mod 4)
N ≡ 4 (mod 5)
We will solve this system using the method of successive substitutions or the Chinese Remainder Theorem.

Step 3: Solve the system of congruences
From the first congruence, N = 4k + 1. Substitute this into the second congruence:

4k + 1 ≡ 4 (mod 5)

Simplifying:

4k ≡ 3 (mod 5)

Now, we solve for k. We need to find the multiplicative inverse of 4 modulo 5. The inverse of 4 modulo 5 is 4 because:

4 × 4 = 16 ≡ 1 (mod 5)

Multiply both sides of the congruence 4k ≡ 3 (mod 5) by 4:

16k ≡ 12 (mod 5)

Simplifying:

k ≡ 2 (mod 5)

Thus, k = 5n + 2 for some integer n.

Step 4: Substitute k back into the expression for N
Now, substitute k = 5n + 2 into the equation N = 4k + 1:

N = 4(5n + 2) + 1 N = 20n + 9

Thus, N = 20n + 9, which means N leaves a remainder of 9 when divided by 20.

Step 5: Find the remainders when N is divided by 5 and 4
Now, we need to find the remainders when N = 20n + 9 is divided by 5 and 4.

When N is divided by 5: 20n + 9 ≡ 9 (mod 5) 9 ≡ 4 (mod 5) So, the remainder when N is divided by 5 is 4.

When N is divided by 4: 20n + 9 ≡ 9 (mod 4) 9 ≡ 1 (mod 4) So, the remainder when N is divided by 4 is 1.

Step 6: Conclusion
Therefore, when the number is successively divided by 5 and 4, the respective remainders are 4 and 1.

The correct answer is D. 4, 1.