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  4. Consider the number N =224-1. Number of two digit divisors of N, is

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Q. Consider the number $N =2^{24}-1$.
Number of two digit divisors of $N$, is

Permutations and Combinations

Solution:

$2^{24}-1=\left(2^{12}+1\right)\left(2^{12}-1\right)=\left(2^{12}+1\right)\left(2^6+1\right)\left(2^6-1\right)$
$=4097 \cdot 65 \cdot 63=241 \cdot 17 \cdot 5 \cdot 13 \cdot 7 \cdot 3^2$
Number of two digit divisior $13 ; 17 ; 21 ; 39 ; 15 ; 51 ; 63 ; 45 ; 91 ; 35 ; 65 ; 85 \rightarrow 12$ divisors.