Its all about numbers

Number Theory Level pending

Find the smallest positive integer n n such that, if you put the number 2 on the left and the number 1 on the right, the new number is equal to 33 n 33n .


The answer is 87.

Sathvik Acharya by Sathvik Acharya , 17, India

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2 solutions

Daniel Branscombe
Oct 20, 2017

we have

33n=10n+1+2*10^k

23n=2 * 10^k+1

So we start by looking for the smallest integer k that results in 2*10^k+1 being divisible by 23. Simple trial and error quickly gives this to be k=3 which gives n=87.

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I generally don't code in Python though. 

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i = 1
while 1:
  j = str(i)
  j = '2' + j
  j += '1'
  j = int(j)
  if j == 33 * i:
    print(i)
    break
  i += 1

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