If N is the number N=7a+4=8b+5=9c+6, where a, b and c are the results of division and are positive integers.
So (N-4)/7=a, (N-5)/8=b, (N-6)/9=c. 7a-8b=1; 7a-9c=2; 8b-9c=1.
Take the first equation: 7a-8b=1, 7a=8b+1. By substituting b=1, 2, 3, ... we can find out values of b where the equation is satisfied. When b=6, a=7, for example. Also, b=13, 20, 27, etc. and a=15, 23, 31, etc. So we have a relationship between a and b. Similarly, 7a=9c+2, and c=6, 13, 20, 27, ... and a=8, 17, 26, 35, ... We have a relationship between a and c. The relationship between b and c is shown by 8b=9c+1: c=7, 15, 23, ... and b=8, 17, 26, ...
Looking at the series for a with b: 7, 15, 23, 31, 39, 47, 55, 63, 71, 79, ...
Looking at the series for a with c: 8, 17, 26, 35, 44, 53, 62, 71, 80, ...
We can see 71 is in both series. We can find out the values of b and c for a=71. 8b=7a-1=496, b=62; 9c=7a-2=495, c=55. N=497+4=496+5=495+6=501.
Another approach is to multiply: 7*8*9=504. They all go into this with zero remainder. We can subtract 3 or add 6 to this number so that the 9's remainder is 6, giving us 501 or 510; we can subtract 3 or add 5 so that the 8's remainder is 5, giving us 501 or 509; we can subtract 3 or add 4 so that the 7's remainder is 4, giving us 501 or 508. Since 501 is the common number, then the number we are looking for is 501.
CHECK: 501/7=71r4; 501/8=62r5; 501/9=55r6.