Lets say there are x diamonds, now these diamonds are exactly divisible by 7.

and

x = 1 + N1*2;

x = 1 + N2*3;

x = 1 + N3*4;

x = 1 + N4*5;

x = 1 + N5*6;

x = N6*7;

where N1, N2, N3, N4 and N5 are integers.

From above we can also say

N1*2 = N2*3 = N3*4= N4*5 = N5*6 = y

Now y should be divisible by 2, 3, 4, 5 and 6, its nothing but common multiple of all.

LCM of 2, 3, 2*2, 5, 2*3 => 2*3*2*5 => 60

at the same time common multiple + 1 should be divisible by 7 as well.

60 + 1 is not divisible by 7

120(60*2) + 1 is not divisible by 7

180(60*3) + 1 is not divisible by 7

240(60*4) + 1 is not divisible by 7

300(60*5) + 1 is divisible by 7

Thus they must have stolen minimum 301 diamonds.