A number is divided successively by 4, then 5, then 6 and the remainders are 2, 3 and 4 respectively. Find the least such number.

Given data

• Successive division remainders: 2 (by 4), then 3 (by 5), then 4 (by 6).

Concept / Approach

• Back-substitute from the last quotient to construct N explicitly.

Step-by-step calculation

Let after first division by 4: N = 4a + 2.Then a ÷ 5 leaves remainder 3 ⇒ a = 5b + 3.Then b ÷ 6 leaves remainder 4 ⇒ b = 6c + 4.Thus N = 4(5(6c + 4) + 3) + 2 = 4(30c + 23) + 2 = 120c + 94.Least positive (c = 0): N = 94.

Verification

• 94 ÷ 4 ⇒ q = 23, r = 2
• 23 ÷ 5 ⇒ q = 4, r = 3
• 4 ÷ 6 ⇒ q = 0, r = 4

Final Answer

Least number = 94.