7 digit number

If 3c54d10 is divisible by 330, what is the sum of c and d?

Result

s1 =  8
s2 =  5
s3 =  11

Solution:

3c54d10 mod  330=0 3c54d1 mod  33=0 33=3⋅ 11  s=c+d 0<=s<=9+9 0<=s<=18  3+c+5+4+d+1 mod  3=0 3+5+4+1+smod  3=0  13+smod  3=0 13=3⋅ 4+1 (1+s)mod  3=0 s=2,5,8,11,15,18   x1=3054810 y1=x1/330=3054810/330=9257 s1=0+8=83c 54d 10 \ \mod 330=0 \ \\ 3c 54d 1 \ \mod 33=0 \ \\ 33=3 \cdot \ 11 \ \\ \ \\ s=c+d \ \\ 0 <=s <=9+9 \ \\ 0 <=s <=18 \ \\ \ \\ 3+c+5+4+d+1 \ \mod 3=0 \ \\ 3+5+4+1+s \mod 3=0 \ \\ \ \\ 13 + s \mod 3=0 \ \\ 13=3 \cdot \ 4 + 1 \ \\ (1+s) \mod 3=0 \ \\ s=2,5,8,11,15,18 \ \\ \ \\ \ \\ x_{1}=3054810 \ \\ y_{1}=x_{1} / 330=3054810 / 330=9257 \ \\ s_{1}=0+8=8
 x2=3454110 y2=x2/330=3454110/330=10467 s2=4+1=5 \ \\ x_{2}=3454110 \ \\ y_{2}=x_{2} / 330=3454110 / 330=10467 \ \\ s_{2}=4+1=5
 x3=3754410 y3=x3/330=3754410/330=11377 s3=7+4=11 \ \\ x_{3}=3754410 \ \\ y_{3}=x_{3} / 330=3754410 / 330=11377 \ \\ s_{3}=7+4=11

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