Explanation:

Let’s break down the problem:

- We need the least number N such that:

N = 10 (mod 15)

N = 20 (mod 25)

N = 30 (mod 35)

N = 35 (mod 40)

- Notice a pattern:

The remainder is always "divisor minus 5".

So, N+5 is divisible by 15, 25, 35, 40.

- Find LCM of 15, 25, 35, 40:

15 = 3×5

25 = 5×5

35 = 5×7

40 = 2³×5

LCM = 2³×3×5²×7 = 4200

- Therefore, N + 5 = 4200, so N = 4200 - 5 = 4195

Let’s verify the options:

- Option 1: 4210

4210 + 5 = 4215, not a multiple of 4200

- Option 2: 4200

4200 + 5 = 4205, not a multiple of 4200

- Option 3: 4205

4205 + 5 = 4210, not a multiple of 4200

- Option 4: 4195

4195 + 5 = 4200, which IS a multiple of 4200

So, the correct answer is:

Option:4, 4195