7 digit number

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

Correct answer:

s₁ =  8
s₂ =  5
s₃ =  11

Step-by-step explanation:

$$\begin{gathered} 3c54d10\mathrm{m}\mathrm{o}\mathrm{d}330=0 \\ 3c54d1\mathrm{m}\mathrm{o}\mathrm{d}33=0 \\ 33=3\cdot 11 \\ s=c+d \\ 0<=s<=9+9 \\ 0<=s<=18 \\ 3+c+5+4+d+1\mathrm{m}\mathrm{o}\mathrm{d}3=0 \\ 3+5+4+1+s\mathrm{m}\mathrm{o}\mathrm{d}3=0 \\ 13+s\mathrm{m}\mathrm{o}\mathrm{d}3=0 \\ 13=3\cdot 4+1 \\ (1+s)\mathrm{m}\mathrm{o}\mathrm{d}3=0 \\ s=2,5,8,11,15,18 \\ {x}_1=3054810 \\ {y}_1=x_1\mathrm{/}330=3054810\mathrm{/}330=9257 \\ {s}_1=0+8=8 \end{gathered}$$

$$\begin{gathered} x_2=3454110 \\ {y}_2=x_2\mathrm{/}330=3454110\mathrm{/}330=10467 \\ {s}_2=4+1=5 \end{gathered}$$

$$\begin{gathered} x_3=3754410 \\ {y}_3=x_3\mathrm{/}330=3754410\mathrm{/}330=11377 \\ {s}_3=7+4=11 \end{gathered}$$

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