M22-27

Official Solution:

If \(f(x)\) is defined as the largest integer \(n\) such that \(x\) is divisible by \(2^n\), which of the following numbers is the greatest?

A. \(f(24)\)
B. \(f(42)\)
C. \(f(62)\)
D. \(f(76)\)
E. \(f(84)\)


Often, the most challenging part is rephrasing a question to fully grasp what it's genuinely asking.

Let's consider an integer \(x\). It contains some power of 2 in its prime factorization (\(2^n\)), and \(f(x)\) represents the value of this \(n\). Essentially, \(f(x)\) is the exponent of 2 in the prime factorization of \(x\).

For instance, if \(x\) equals 40, then \(f(x)=3\). Why? Because the largest integer \(n\) for which 40 is divisible by \(2^n\) is 3: \(\frac{40}{2^3}=5\), or in other words, \(40=2^3*5\) - the power of 2 in the prime factorization of 40 is 3.

So, to answer the question, all we need to do is factorize all options and determine which one contains the highest power of 2.

A. \(f(24)\). Factorize 24: \(24 = 2^3*3\), so \(f(24) = 3\)

B. \(f(42)\). Factorize 42: \(42 = 2*21\), so \(f(42) = 1\)

C. \(f(62)\). Factorize 62: \(62 = 2*31\), so \(f(62) = 1\)

D. \(f(76)\). Factorize 76: \(76 = 2^2*19\), so \(f(76) = 2\)

E. \(f(84)\). Factorize 84: \(84 = 2^2*21\), so \(f(84) = 2\)


Answer: A