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2 ways to prove it (Posted on 2017-10-31) Difficulty: 2 of 5
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    
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Solution my offered proofs... are they really the same Comment 2 of 2 |
One way:

10^n cycles through 10, 19, 1 ... mod 27, a cycle of length 3

18*n cycles through 18, 9, 0 ... mod 27, again a cycle of 3

The total, mod 27 is always 1, so when you subtract 1, the value is 0 mod 27, or in other words the number is divisible by 27.

Is this a different proof? ... proof by induction:

10^1 is congruent to 10 mod 27

18*1 is congruent to 18 mod 27

therefore, for n=1 10^n + 18*n - 1 = 0 mod 27

10^(n+1) - 10^n is 9 mod 27

18*(n+1) - 18*n is -9 mod 27

therefore if 10^n + 18*n - 1 = 0 mod 27 for n=k, then it is true for n = k+1.

  Posted by Charlie on 2017-10-31 13:58:26
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