Sum of integers!

Number Theory Level 2

What is the sum of all positive integer values of n n such that n + 18 n \frac{n+18}{n} is an integer?


The answer is 39.

Swapnil Das by Swapnil Das , 20, India

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1 solution

Aditya Chauhan
Aug 23, 2015

n + 18 n \dfrac{n+18}{n}

= 1 + 18 n =1+\dfrac{18}{n}

The given expression will be an integer only when n is a factor of 18.

The factors of 18 18 are 1 , 2 , 3 , 6 , 9 , 18 1,2,3,6,9,18 .

So Sum = 1 + 2 + 3 + 6 + 9 + 18 = 39 =1+2+3+6+9+18 =\boxed{39}

7 Helpful 0 Interesting 0 Brilliant 0 Confused

Good solution! Here is a faster way to add the divisors:

Since the prime factorization of 18 is 18 = 2 ⋅ 3 2 18 = 2\cdot 3^2 , the sum of its divisors is ( 1 + 2 ) ⋅ ( 1 + 3 + 9 ) = 3 × 13 = 39. (1 + 2)\cdot (1 + 3 + 9) = 3\times 13 = 39.

Arjen Vreugdenhil - 5 years, 8 months ago

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