Question

Find the least number which when divided by 20, 25, 35 and 40 leaves remainders 14, 19, 29 and 34 respectively.

Answer 1

Given: N ≡ 14 (mod 20), N ≡ 19 (mod 25), N ≡ 29 (mod 35), N ≡ 34 (mod 40).

Step-wise calculation:

1. Compute modulus minus remainder for each:

20 – 14 = 6

25 – 19 = 6

35 – 29 = 6

40 – 34 = 6

Hence, N + 6 is divisible by 20, 25, 35 and 40.

2. Find LCM(20, 25, 35, 40): 

20 = 2² × 5 

25 = 5²

35 = 5 × 7

40 = 2³ × 5 

LCM = 2³ × 5² × 7 

= 8 × 25 × 7 

= 1400

3. Smallest positive choice:

N + 6 = 1400

⇒ N = 1400 – 6

⇒ N = 1394

4. Quick check:

1394 ÷ 20 = 69 R 14

1394 ÷ 25 = 55 R 19

1394 ÷ 35 = 39 R 29

1394 ÷ 40 = 34 R 34

The least such number is 1394.