Proof Problem Of The Day - Digits and Sums!

For a positive integer NN, let S(N)S(N) be the sum of digits in the decimal representation of NN. Let F(N)F(N) be the sum of all the numbers obtained by erasing one or more numbers from the right end of NN.

Prove that S(N)+9F(N)=NS(N)+9F(N)=N


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Note by Mursalin Habib
4 years, 10 months ago

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At first let N=k=0nak10kN=\displaystyle\sum_{k=0}^n a_k 10^k. We have S(N)=k=0nakS(N)=\displaystyle \sum_{k=0}^n a_k. Now we calculate and simplify F(N)F(N). A little observation is required to find the compact form of F(N)F(N)

F(N)=i=1nj=inaj10ji=i=1nj=0i1ai10j=i=1n(aij=0i110j)=i=1nai10i19.F(N) = \sum_{i=1}^n \sum_{j=i}^n a_j 10^{j-i}=\sum_{i=1}^{n} \sum_{j=0}^{i-1} a_{i} 10^j=\sum_{i=1}^n\left(a_{i}\sum_{j=0}^{i-1} 10^j\right)=\sum_{i=1}^n a_{i}\dfrac{10^i-1}{9}.

The rest is straight. Using the last form of F(N)F(N) we have

9F(N)=i=1nai(10i1)=i=1n(ai10iai)=i=1nai10ii=1nai=NS(N)9F(N)=\sum_{i=1}^n a_i\left(10^i -1\right)=\sum_{i=1}^n \left(a_i10^i-a_i\right)=\sum_{i=1}^n a_i10^i -\sum_{i=1}^n a_i =N-S(N)

and the result follows. \square


NOTE: If you're wondering how I reformed the sum of F(N)F(N) in the third step above, say N=3265N=3265. Then clearly

F(3625)=326+32+3.F(3625)=326+32+3.

This is the form of the first sum. Now break further into

F(3625)=326+32+3=(300+20+6)+(30+2)+3=(300+30+3)+(20+2)+6=3(102+101+100)+2(101+100)+6100.\begin{aligned} F(3625) &=&326+32+3 \\ &=&(300+20+6)+(30+2)+3 \\ &=&(300+30+3)+(20+2)+6 \\ &=&3(10^2+10^1+10^0)+2(10^1+10^0)+6\cdot 10^0. \end{aligned}

Observe the number of occurrences of the powers of 1010 for the nnth digit and generalize to get the second sum.

Jubayer Nirjhor - 4 years, 10 months ago

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Great job!

Another way of doing it is inducting on the number of digits of NN.

Clearly the statement is true for any one digit-number.

Assume that the statement is true for any kk-digit number.

Now for any (k+1)(k+1)- digit number bb, let b=10a+cb=10a+c where aa is a kk-digit number and cc is a one-digit number.

Notice that, S(b)=S(a)+cS(b)=S(a)+c and F(b)=F(a)+aF(b)=F(a)+a

So, S(b)+9F(b)=S(a)+c+9F(a)+9aS(b)+9F(b)=S(a)+c+9F(a)+9a.

From the induction hypothesis, S(a)+9F(a)=aS(a)+9F(a)=a. Substituting that, we get,

S(b)+9F(b)=10a+c=bS(b)+9F(b)=10a+c=b.

So the result holds for all positive integers NN.

Mursalin Habib - 4 years, 10 months ago

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BTW, someone's downvoting every damn comment he's passing by. -_-

Jubayer Nirjhor - 4 years, 10 months ago

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@Jubayer Nirjhor I don't mind being downvoted.

Mursalin Habib - 4 years, 10 months ago

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@Mursalin Habib Neither do I. But I do mind being spammed.

Jubayer Nirjhor - 4 years, 10 months ago

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At first it seems rather tedious. However, if we prove by induction on the number of digits the calculations will magically disappear!

When number of digits is 1, F(N)=0,S(N)=nF(N)=0, S(N)=n, done.

Suppose it was true for all kk digit numbers. Consider a (k+1)(k+1) digit number XX with last digit mm. Let YY be the kk digit number without mm. Then X=10Y+mX=10Y+m. Hence S(X)+9F(X)=S(Y)+m+9F(Y)+9YS(X)+9F(X)=S(Y)+m+9F(Y)+9Y (since we missed out YY when calcualting F(Y)F(Y)).

But by inductive hypothesis, S(Y)+9F(Y)=YS(Y)+9F(Y)=Y, hence S(X)+9F(X)=Y+m+9Y=10Y+m=XS(X)+9F(X)=Y+m+9Y=10Y+m=X and we are done.

Joel Tan - 4 years, 9 months ago

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Sorry, I did not see Mursalin's comment below. I just realized it was a repetition.

Joel Tan - 4 years, 9 months ago

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Interesting.

Shubham Agrawal - 4 years, 8 months ago

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