A number leaves remainders 1 and then 4 when divided successively by 4 and 5. If instead it is divided successively by 5 and then 4, what are the respective remainders?

Given data

• Successive division by 4 then 5 yields remainders: 1 (mod 4), then 4 (mod 5).
• Find the remainders if the order is reversed: first divide by 5, then divide the quotient by 4.

Concept / Approach

• Translate successive remainders into algebra and back substitute.
• Then recompute remainders when the order of the divisors is reversed.

Step-by-step calculation

Let the original number be N.Since division by 4 leaves remainder 1: N = 4a + 1 for some integer a.Now divide the quotient a by 5 and get remainder 4: a = 5b + 4 for some integer b.Therefore N = 4(5b + 4) + 1 = 20b + 17.

Reversed order: first by 5, then by 4

First remainder (mod 5): N mod 5 = (20b + 17) mod 5 = 0 + 2 = 2.Quotient after dividing by 5: Q = (N − 2)/5 = (20b + 15)/5 = 4b + 3.Second remainder (mod 4): Q mod 4 = (4b + 3) mod 4 = 3.

Verification / Alternative

Choose b = 0 (the least case): N = 20·0 + 17 = 17.Check original order: 17 ÷ 4 ⇒ q = 4, r = 1; then 4 ÷ 5 ⇒ q = 0, r = 4 ✓Check reversed order: 17 ÷ 5 ⇒ q = 3, r = 2; then 3 ÷ 4 ⇒ q = 0, r = 3 ✓

Common pitfalls

• Confusing simultaneous congruences with successive division. Here the second remainder applies to the quotient from the first division.
• Trying to combine the divisors into a single modulus; that does not model successive operations correctly.

Final Answer

On reversing the order, the remainders are 2 and 3.