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 $k$ is equal to

• A. $2$
• B. $3$
• C. $4$
• D. $8$
• E. $18$

Answer 1

Answer: (B)

$3$

Given,$2^k$ is divisor of $72$

Finding prime divisors of $72$

$\Rightarrow 72=2^3{\times 3}^2$

Now it is given,$2^k$, i.e $2^3=8$ is a divisor of $72$

Therefore, taking $k=3$

Also given that $a^{k+1}$, i.e $2^{k+1}$ is not a divisor

$2^{3+1}=2^4=16$

Now $\frac{72}{16}=\frac{9}{2}$

Hence $2^{k+1}$ is not a divisor of $72$.

Considering all points, $k=3$