Question

For the positive integers a, b and k, ${{a}^{k}}\parallel b$ means that ${{a}^{k}}$ is a divisor of b, but ${{a}^{k+1}}$ is not a divisor of b. If k is a positive integer and ${{2}^{k}}\parallel 72$, then find the value of k?
(a) 2
(b) 3
(c) 4
(d) 8
(e) 18

Answer 1

Hint: We start solving the problem by factoring the numbers up to the limit that it cannot be further factorized by any prime number. We then check what is the maximum power of 2 that can divide 72 completely, which lets us know that the next power of 2 cannot divide 72. We then compare the obtained power of 2 with ${{2}^{k}}$ to find the value of k.

Complete step-by-step solution
According to the problem, we are given that ${{a}^{k}}\parallel b$ means that ${{a}^{k}}$ is a divisor of b, but ${{a}^{k+1}}$ is not a divisor of b for the positive integers a, b and k. We need to find the value of k, if it is given ${{2}^{k}}\parallel 72$.
Let us first factorize the number 72 and find what will be the maximum value of ${{2}^{k}}$ that can divide 72 completely.
So, we know that $72=2\times 36$.
$\Rightarrow 72=2\times 2\times 18$.
$\Rightarrow 72=2\times 2\times 2\times 9$
$\Rightarrow 72=2\times 2\times 2\times 3\times 3$.
$\Rightarrow 72={{2}^{3}}\times {{3}^{2}}$ ---(1).
We can see that ${{2}^{3}}$ completely divides the number 72 and is the maximum power of 2 that can divide 72 completely.
So, ${{2}^{4}}=16$ cannot divide 72.
So, comparing ${{2}^{k}}$ with ${{2}^{3}}$, we get the value of k as 3.
∴ The value of k is 3.
The correct option for the given problem is (b).

Note: We can solve this problem by making use of trial and error methods for the value of k. starting from 1 till the value of k that ${{2}^{k}}$ cannot divide 72 completely. We should not make calculation mistakes while factorizing 72. Whenever we get this type of problem, factorization will be a good tool to get the required result. Similarly, we expect problems to find the value of k if it is given ${{3}^{k}}\parallel 72$.