An integer is said to have a perfect square factor if there is a prime number such that divides . For example, 12 has a perfect square factor because but 14 has none because .
is a composite number which has no perfect square factor ( ) and neither does the sum of its factors: .
What is the previous number to share those three properties ?
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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 = 1 9 9 4