Math Puzzles

# Can you solve this number puzzle?

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## Problem Statement

• N is a positive integer.
• If we add N and the sum of its digits, then we get 313.

What are the possible values of N?

## Solution

How to solve the puzzle?

Let's start with the largest two-digit number i.e. 99. Let’s add 99 and the sum of its digits:

 99 + 9 + 9 = 117

Therefore, N must be a 3-digit number. Also, N cannot exceed 313.

Next, the maximum sum of digits of a 3-digit number is 27.

Therefore, N must at least be 313-27 = 286.

Hence,

 286 < N < 313

We can say that N is either of the form 28p or 29p or 30p or 31p (where p is the unit’s digit).

For the form 28p:

28p + (2 + 8 + p) = 313

280 + 10 + 2p = 313

This is not possible since the LHS is clearly an even number, while 313 is odd.

For the form 29p:

29p + (2 + 9 + p) = 313

290 + 11 + 2p = 313

290 + 2p = 313-11 = 302

2p = 302 – 290 = 12

 p = 12/2 = 6

Therefore, the first possible value of N is 296.

For the form 30p:

30p + (3 + 0 + p) = 313

300 + 3 + 2p = 313

2p = 313 – 300 – 3 = 10

 p = 10/2 = 5

Thus, the second possible value of N is 305.

For the form 31p:

31p + (3 + 1 + p) = 313

310 + 4 + 2p = 313

314 + 2p = 313

This is not possible since the LHS is clearly greater than 313 (314 is greater than 313 and N has to be a positive integer).

So, we have two possible values of N, 296, and 305.

## Conclusion

Hope you liked the challenge! At Cuemath, we regularly present challenging and stimulating math puzzles to our students so that their logical thinking and problem-solving abilities are enhanced.

**This puzzle appeared in the book – The Ultimate Mathematical Challenge Over 365 puzzles to test your wits and excite your mind, written by Marcus du Sautoy, a Professor of Mathematics and Simonyi Professor for the Public Understanding of Science at the University of Oxford.

Cuemath, a student-friendly mathematics platform, conducts regular Online Live Classes for academics and skill-development, and their Mental Math App, on both iOS and Android, is a one-stop solution for kids to develop multiple skills.

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