Steve has recently got his annual exam result.He has got upper than 690 out of 750.His obtained marks has odd number of factors.What is his obtained marks?
2026-04-02 09:28:28.1775122108
Specific Annual Examination Marks
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Hint: The positive integer $n$ has an odd number of factors if and only if $n$ is a perfect square.
Remark: You have probably seen a proof of the above fact. If you have not, if $uv=n$ and $u\ne v$, call the pair $\{u,v\}$ a couple, or, to be more modern, partners.
If $n$ is not a perfect square, the divisors of $n$ break up into partner pairs, so there is an even number of divisors. If $n$ is a perfect square $m^2$, then the divisor $m$ of $n$ remains unpartnered, so we get the partnered people (even total number) plus the unpartnered one, so the number of divisors is odd.
Work out the details for say $n=12$ and $n=36$. For each one, write down the couples. (For $36$, the divisor $6$ remains uncoupled.) Now you will know the why of the result forever.