why only perfect squared number have an unpair number of divisors
In: Mathematics
Because if you divide a number by its divisor, you get another divisor. Divisors are naturally paired up like this. The only way you can end up with odd number of divisors is if a divisor pairs up with itself, which means the original number is square of this divisor.
Let’s take an example; say 12 which can be expressed as 1×12 (trivial), 2×6 and 3×4.
You’ll note that kind of by the definition of divisors, they always come in pairs. A number (a) is a divisor of 12 if 12/(a) = (b) where (b) is an integer. But simple algebra then also means that if (a) is a divisor (b) is also because 12/(b) = (a). This is obvious if we look at our example, 3 is a divisor of 12 because 12/3 = 4, but also 4 must be a divisor because 12/4 = 3. So divisors always come in pairs and there in principle should always be an even number of them.
The interesting part here is there’s one situation we glossed over which is where (a) = (b). For example, 4 = 2×2. Here we can see 2 is a divisor, but because it’s multiplied by itself it doesn’t come with another new divisor in a pair. This only happens with square numbers (by defintion as you can write them as (a)x(a)), and hence square numbers are the only ones to have an odd number of divisors.
ELI5 what is an unpair of divisors…?
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