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Algebra Difficulty 2.3 Junior Find the answer

In the subtraction shown, K,L,MK, L, M, and NN are digits. What is the value of K+L+M+NK+L+M+N?\text{?}6K0LM9N42011\begin{array}{r}6 K 0 L \\ -\quad M 9 N 4 \\ \hline 2011\end{array}

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

We work from right to left as we would if doing this calculation by hand. In the units column, we have L4L-4 giving 1. Thus, L=5L=5. (There is no borrowing required.) In the tens column, we have 0N0-N giving 1. Since 1 is larger than 0, we must borrow from the hundreds column. Thus, 10N10-N gives 1, which means N=9N=9. In the hundreds column, we have K9K-9 but we have already borrowed 1 from KK, so we have (K1)9(K-1)-9 giving 0. Therefore, we must be subtracting 9 from 9, which means that KK should be 10, which is not possible. We can conclude, though, that K=0K=0 and that we have borrowed from the 6. In the thousands column, we have 5M=25-M=2 or M=3M=3. This gives 60053994=20116005-3994=2011, which is correct. Finally, K+L+M+N=0+5+3+9=17K+L+M+N=0+5+3+9=17.

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