SOLUTION: Find the greatest number of four digits which when increased by 1 is exactly divisible by 2,3 ,4,5,6 and 7.

For a positive integer to be divisible by those
it must divisible by their Least common multiple:

We find the least common multiple of 2,3,4,5,6, and 7

Break each into primes if it is not prime

2=2
3=3
4=2*2
5=5
6=2*3
7=7

so LCM = 2*2*3*5*7 = 420 

Every multiple of 420 is divisible by all those above.
Every multiple of 420 can be written 420n, where
n is an integer. The largest 4-digit integer is
9999. Therefore: 






Since n is an integer


Therefore the largest multiple of 420
which has 4 digits is 420*23 or 9660.

However the problem asked for the largest digit
that when increased by 1 will give 9660. Obviously
this is 1 less than 9660, or 9659.

Answer = 9659.

Edwin