Just Simplify That Power Tower!

Number Theory Level 1

$\large \text{lcm} (2^1 ,2^2 ,2^3,\ldots, 2^n) = 2^{2^{2^2}}$

What is the value of $n$ satisfying the equation above?

Notation : $\text{lcm}(\cdot)$ denotes the lowest common multiple function.

The answer is 16.

3 solutions

A Former Brilliant Member
May 27, 2016

Relevant wiki: Lowest Common Multiple

$2^{a} \vert 2^{b} , b \ge a$
$\therefore \text{lcm}(2,2^{2}, 2^{3},\ldots, 2^{n}) = 2^{n}$
$2^{n} = 2^{2^{2^{2}}}$
$2^{n} = 2^{16} \to n = 16$

Yup! This is correct! Another way to solve this is by induction . Thank you!

Pi Han Goh - 5 years ago

2^2^2^2 = 2^8

suman kumar - 5 years ago

Nope. Read up how are exponent towers evaluated?

Pi Han Goh - 5 years ago

thank you. got it :)

suman kumar - 5 years ago

Lance Fernando
Jun 4, 2016

Exponential Tower: 2^2 = 4^2 = 16

2^2 = 4^2 is wrong.

Pi Han Goh - 5 years ago

I started at the top, that's why. 2^2 = 4. 2^4 = 4^2. So still, the answer is the same. :D

Lance Fernando - 5 years ago

Ashish Menon
May 26, 2016

${{2^2}^2}^2 = {2^2}^4 = 2^{16}$ .
Now, when we take Lcm of ascending powers of two starting from $2^1$ , one two alternatively comes out common and one term aletrnatively becomes $1$ . So, the minimum value of $n$ should be $16$ .

If $n = 16$ , then in each step of taking out the LCM, one $2$ gets common and it continues $16$ times giving us the LCM $2^{16}$ which is what we want.

Just Simplify That Power Tower!

The answer is 16.

3 solutions

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