Question

Which of the following statements is true ${2^{16}} - 1$ is divisible by
A. $11$
B. $13$
C. $17$
D. $19$

Answer 1

Hint: In the above question we have the expression ${2^{16}} - 1$ , so we will first break down this expression and then the number we get after solving, we will check by which of the given numbers it is completely divisible. We can break ${2^{16}}$ into the simpler terms, it can be written as ${({2^4})^4}$ , here we have broke down the exponent i.e. $16 = 4 \times 4$.

Complete step by step answer:
Here we have ${2^{16}} - 1$.So we can write
${2^{16}}$ as ${({2^4})^4} = {(16)^4}$ and we can also write $1$ as ${1^4}$ .
We can write the expression as
${(16)^4} - {(1)^4}$
Now here we have the expression in the form of
${a^4} - {b^4}$ , where $a = 16,b = 1$.
We will now simplify the algebraic expression, we can write ${a^4}$ as the square of ${a^2}$ i.e.
${a^4} = {({a^2})^2}$
Similarly we can write ${b^4}$ as the square of ${b^2}$ i.e.
${b^4} = {({b^2})^2}$
Thus the expression becomes:
${a^4} - {b^4} = {({a^2})^2} - {({b^2})^2}$

Now we know the algebraic formula of difference of square, which is
${x^2} - {y^2} = (x + y)(x - y)$
So by comparing this from the above expression we have
$x = {a^2}$ and $y = {b^2}$
So we can write
 ${a^4} - {b^4} = ({a^2} + {b^2})({a^2} - {b^2})$
Again we can apply the difference formula on the second part i.e. $({a^2} - {b^2})$
By comparing this time we have:
$x = a$ and $y = b$
So we can write the new expression i.e.
${a^4} - {b^4} = ({a^2} + {b^2})(a + b)(a - b)$
This is the algebraic formula of the form:
${a^4} - {b^4}$

Now we can apply the algebraic formula i.e.
${a^4} - {b^4} = (a + b)(a - b)({a^2} + {b^2})$
By comparing from the above expression we have
$a = 16,b = 1$.
By putting the values in the expression, it gives:
$(16 - 1)(16 + 1)({16^2} + {1^2})$.
On further simplifying we have:
$15 \times 17 \times (256 + 1) = 15 \times 17 \times 257$
We will now multiply all the three numbers and the number required is $65535$ .
Now we can see say that the factors of $65535$ is
$15,17,257$
Or we can say,
$5,3,17,257$
So it will be fully divisible by all the above three numbers i.e. its multiples.
So from the above given options ${2^{16}} - 1$ is divisible by $17$ .

Hence the correct option is C.

Note:We should note that in the above solution we have written ${({2^4})^4}$ as ${(16)^4}$ i.e. ${2^4} = 2 \times 2 \times 2 \times 2$, it gives the value $16$ . Also we should remember the perfect squares and their numbers as the square of $16$ is $16 \times 16 = 256$ . We can also solve the above question directly i.e. ${2^{16}} - 1 = 65536 - 1$ .
It gives the value $65535$ and then we write down its factors i.e. $3 \times 5 \times 17 \times 257$.