Teachers shouldn't have ugly handwriting

Number Theory Level pending

My math teacher wrote the number 7 6 □ \large 7^{6^{\square}} , the rightmost number was omitted because I couldn't decipher his ugly handwriting.

I couldn't for the life of me figure out what should be the missing number, but I do recall that the teacher (truthfully) told us that this number has 7776 proper divisors.

What is this missing number?


The answer is 5.

Pi Han Goh by Pi Han Goh , 30, Malaysia

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Michael Huang
Dec 3, 2016

Since 7 7 is the prime number, it is counted as one proper divisor. Then, we need to find the value of □ \square , such that 6 □ = 7776 6^{\square} = 7776 which shows that there are 7776 of 7 ’s 7\text{'s} . It's clear that by taking logarithm for both sides, log ⁡ ( 6 □ ) = log ⁡ ( 7776 ) □ log ⁡ ( 6 ) = log ⁡ ( 7776 ) □ = log ⁡ ( 7776 ) log ⁡ ( 6 ) = log ⁡ ( 6 5 ) log ⁡ ( 6 ) = 5 \begin{array}{rl} \log\left(6^{\square}\right) &= \log\left(7776\right)\\ \square \log\left(6\right) &= \log\left(7776\right)\\ \square &= \dfrac{\log\left(7776\right)}{\log\left(6\right)}\\ &= \dfrac{\log\left(6^5\right)}{\log\left(6\right)}\\ &= \boxed{5} \end{array}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...