What to expect from 2018?

An integer n n is said to have a perfect square factor if there is a prime number p p such that p 2 p^2 divides n n . For example, 12 has a perfect square factor because 12 = 2 2 × 3 12 = 2^{\color{#D61F06}2} \times 3 but 14 has none because 14 = 2 × 7 14 = 2 \times 7 .

2018 2018 is a composite number which has no perfect square factor ( 2018 = 2 × 1009 2018 = 2 \times 1009 ) and neither does the sum of its factors: 1 + 2 + 1009 + 2018 = 3030 = 2 × 3 × 5 × 101 1+2+1009+2018 = 3030 = 2 \times 3 \times 5 \times 101 .

What is the previous number to share those three properties ?


The answer is 1994.

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1 solution

Vaibhav Bhatt
Jan 6, 2018

Its easy to see that if 'n' has more than 2 prime factors other than the number 2, then its sum of divisors will be divisible by 4. So 'n' has only one prime factor other than 2, and it is also divisible by 2. So N=2p, and p+1 must not be divisible by 3 and must not have a perfect square factor. Only number before 1009 exhibiting such properties is 997. Hence n = 1994 n=1994

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