<s>[INST] How many numbers? How many numbers from 7, 13, 14, 21, 26, 39 and 91 divide $$(69^{12} − 38^{12})$$?
I am able to get the answer by calculating it.
But is there is any shorter way by which i can solve this? 
Thanks in advance.
 [/INST] You can calculate the expression in question modulo 7, 13, etc.
For example, $69^{12}-38^{12} \mod 7 = (-1)^{12}-3^{12} \mod 7 = 1-2^{6} \mod 7 = 0$.
Another way is to note that $x^{12}-y^{12} = (x^6)^2-(y^6)^2 = (x^6-y^6)(x^6+y^6) = (x^3-y^3)(x^3+y^3)(x^6+y^6) = (x-y)(x^2+xy+y^2)(x^3+y^3)(x^6+y^6)$, and $69^2+69\times38+38^2 = 4761+2622+1444=8827=7\times13\times97$.
Also note that $14=2\times7$, $21=3\times7$, $26=2\times13$, $39=3\times13$ and $91=7\times13$. Yet $69^{12}-38^{12} = 3^{12}\times23^{12}-2^{12}\times19^{12}$ obviously does not divides neither $2$ nor $3$, so you have to check only divisibility by $7$ and $13$ (and conclude the divisibility by $91$ from that).
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