# 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:

$3c 54d 10 \ \mod 330 = 0 \ \\ 3c 54d 1 \ \mod 33 = 0 \ \\ 33 = 3 \cdot \ 11 \ \\ \ \\ s = c+d \ \\ 0 < = s < = 9+9 \ \\ 0 < = s < = 18 \ \\ \ \\ 3+c+5+4+d+1 \ \mod 3 = 0 \ \\ 3+5+4+1+s \mod 3 = 0 \ \\ \ \\ 13 + s \mod 3 = 0 \ \\ 13 = 3 \cdot \ 4 + 1 \ \\ (1+s) \mod 3 = 0 \ \\ s = 2,5,8,11,15,18 \ \\ \ \\ \ \\ x_{ 1 } = 3054810 \ \\ y_{ 1 } = x_{ 1 } / 330 = 3054810 / 330 = 9257 \ \\ s_{ 1 } = 0+8 = 8$
$\ \\ x_{ 2 } = 3454110 \ \\ y_{ 2 } = x_{ 2 } / 330 = 3454110 / 330 = 10467 \ \\ s_{ 2 } = 4+1 = 5$
$\ \\ x_{ 3 } = 3754410 \ \\ y_{ 3 } = x_{ 3 } / 330 = 3754410 / 330 = 11377 \ \\ s_{ 3 } = 7+4 = 11$

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