Problem 23

Find the sum of digits of the least positive integer whose first digit is 7 and which is reduced to a third of its original value when its first digit is transferred to the end.


The answer is 126.

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5 solutions

Noureldin Yosri
Jan 13, 2016

if L is the number of digits of the number then we can write the number as N = 7 1 0 L 1 + x N = 7*10^{L-1} + x for L > 1 clearly L = 1 (N = 7) is not a solution .... the transformation makes N n e w = x 10 + 7 N_{new} = x*10 + 7 so 7 1 0 L 1 + x = 3 ( x 10 + 7 ) 7 1 0 L 1 + x = 30 x + 21 x = 7 1 0 L 1 21 29 x Z + 7 1 0 L 1 21 % 29 ϕ ( 29 ) = 28 g c d ( 29 , 7 ) = 1 1 0 L 1 21 7 1 % 29 1 0 L 1 21 7 ϕ ( 29 ) 1 % 29 1 0 L 1 21 25 % 29 1 0 L 1 3 % 29 7*10^{L-1} + x = 3*(x*10 + 7)\\ 7*10^{L-1} + x = 30*x + 21 \\ x = \frac{7*10^{L-1}-21}{29} \\ x \in Z^+ \\ 7*10^{L-1} \equiv 21 \% 29 \\ \phi (29) = 28 \\ gcd(29,7) = 1 \\ 10^{L-1} \equiv 21*7^{-1} \% 29 \\ 10^{L-1} \equiv 21*7^{ \phi (29) - 1} \% 29 \\ 10^{L-1} \equiv 21*25 \% 29 \\ 10^{L-1} \equiv 3 \% 29

the least L which holds the equation true is 28 from which we get x then we get the answer

Great explanation for how to solve this systematically :)

Calvin Lin Staff - 5 years, 4 months ago
Ivan Koswara
Jan 11, 2016

The number is 7241379310344827586206896551. The following is not rigorous; it's just some kind of intuition. The idea is like this:

  1. The first digit is 7. When it's moved to the back, the result is one-third of it, so the first digit of the result is 2. This first digit is the second digit of the original number.
  2. The first two digits are 72. The second digit of the result, and thus the third digit of the original number, must then be 4 (so that it's one-third of 72...).
  3. The first three digits are 724. The third digit of the result is then 1.

This is repeated until we find a 7 that doesn't have any carry over. (In the fifth step, we will encounter a 7, but without the number itself fully divisible yet (72413 divided by 3 is 24137 with remainder 2), so we cannot stop there.) This gives that number.

There's a better way of solving it. Think about how to set up the algebraic equation, if the number has k k digits.

Calvin Lin Staff - 5 years, 5 months ago

What about 7241379310 ?

Akshat Sharda - 5 years, 5 months ago

Log in to reply

7241379310 / 3 = 2413793103.333 7241379310 / 3 = 2413793103.333\ldots

Ivan Koswara - 5 years, 5 months ago
Soumava Pal
Feb 24, 2016

We follow a simple algorithm after having made an important conclusion: First, since a number P P with 7 at the beginning is thrice of another smaller number s s , it must have '24' or '25' or '26' at the beginning because only these three numbers when multiplied by 3 give a number with 7 at the beginning. Now our algorithm is as follows: We start building the number s s as well as P P from the rightmost place, that is, the unit's place, where we know that there is 7, and at each step we follow the following rule:

A. If the leftmost digit in s s is d d , and there is some 'carrying digit c c above it' (which may be 0) and that in P P is d d' , multiply the leftmost digit d d in s s with 3 and add c c to it.

B. You will get a 2-digit number say 10 t + u 10t+u , where t t is the ten's digit and u u is the unit's digit. We write u u to the left of the d d' in P P , and also write u u to the left of d d in s s .

C. Next we write t t as the carrying digit over the leftmost digit u u in s s , so that we have the transform,

d = u d=u

d = u d'=u

c = t c=t

D. Next we return to step A, unless we have got one of the numbers 24, 25 or 26 as the beginning digits of P P .

I did this by hand, so here are the first few steps.

s = . . . . 7 s=....7 and P = . . . . 1 P=....1

s = . . . . 17 s=....17 and P = . . . . 51 P=....51

s = . . . . 517 s=....517 and P = . . . . 551 P=....551

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and so on.

Finally the number P P will come out to be 7241379310344827586206896551 7241379310344827586206896551 .

P.S. This would make a fine algorithmic question for computer science as well, because brute force search won't work here, due to the sheer magnitude of the number.

Same here exactly.

D. H. Kim - 5 years, 3 months ago
Manit Kapoor
Jan 15, 2016

Use code for solving but logic is the main here

So the answer is 126

Jacob Kirmayer
Jun 30, 2019

I am going to share a tactic that I did and it worked well for this problem, (but you might need a calculator) :

The first digit is 7 and 7/3 = 2.33333 , so the first digit of the number divided by 3 is 2. Thus the first 2 digits are 72. So the first 2 digits are 72. 72/3 = 24. So we know that the first two digit of the number divided by 3 is 24. Thus the first 3 digits of this number are 724.

We repeat this process until it divides perfectly and the last digit is 7. We arrive at 7241379310344827586206896551 after 28 digits of this method.

And the digit sum is 126

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