Find the least value of K for which 7864K3 is divisible by 7.

The given six-digit number is 7864K3.
Let us divide the number by 7 by expanding it or using normal division to find the condition on K:

7
786403+10K
​

Let's divide 786403 by 7:

786403=7×112343+2
So, the remainder when 786403 is divided by 7 is 2.

Thus, the remainder of the full expression is the remainder of:

7
2+10K
​

Let us test the given options for the least value of K:

If K=1: 2+10(1)=12 (Not divisible by 7)

If K=2: 2+10(2)=22 (Not divisible by 7)

If K=4: 2+10(4)=42 (42 is divisible by 7, since 7×6=42)

If K=5: 2+10(5)=52 (Not divisible by 7)

Hence, the least value of K is 4.