perplexus.info :: Just Math : 2 ways to prove it

2 ways to prove it (Posted on 2017-10-31)

For every positive integer n: 10ⁿ+18*n-1 is divisible by 27.

The above statement can be proven by more than one way.

Find at least 2 distinct methods.

No Solution Yet Submitted by Ady TZIDON

Rating: 4.0000 (1 votes)

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Puzzle Solution: Method 1

| Comment 3 of 4 |

10^n + 18*n -1 10^n - 1 18*n

------------------------ = -------------- + ---------

9 9 9

= 11.....11 (n 1s) + 2*n

Now, {11....11(n 1s) +2*n} (mod 3)

= n (mod 3) + 2*n (mod 3)

= 3*n (mod 3)

= 0 (mod 3)

Consequently, 10^n+18*n-1 is divisible by 9*3 = 27

Posted by K Sengupta on 2022-05-14 23:38:14

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