Question

what is the least number that must be added to 2000 so that the sum is divisible exactly 10,12,16and 18 ? do it step by step

Answer 1

The integers divisible by any set of positive integers are the multiples of their LCM

let us first write the factored form of each

10 = 2×5

12 = 2×2×3

16 = 2×2×2×2

18 = 2 x3×3

Now we will find lcm of these numbers

LCM = 2×2×2×2×3×3×5 = 720

The multiples of 720 are divisible by 10,12,16 and 18.

2000/720 = 2.777777...

The least integer greater than that is 3, so 3×720 = 2160 is the least integer greater than 2000 that is divisible by 10,12,16 and 18.

so if we need to find what must be added to 2000 so that the sum is divisible by 10,12,16 and 18, we must subtract 2000 from 2160

2160-2000=160

so we must add 160 to 2000 so that the sum is divisible exactly 10,12,16and 18