Understand the problem
Source of the problem
Elementary Number Theory by David M. Burton
Start with hints
Check the problem out…give its statement a thorough read. Might appear a bit daunting on the first couple of reads. Think for some time, you could be on to something without any help whatsoever !
Okay, now let’s think about what our first thoughts could be, on the problem. It’s definitely about the n in the problem, which acts as our unknown here. Can you somehow try finding the n ? Let’s take the first step in that direction. How could we prime factorize 110 ? That’s easy 110 = 2.5.11. Could you take things from hereon to find more about the n ?
However interestingly the problem says, the number 110. (n^3) has 110 factors. Just as we saw, 110. (n^3) = 2.5.11.(n^3) Now, let’s use some basic number theoretic knowledge here. How many divisors would 110. (n^3) have then ? If n=1 Clearly it would have, (1+1). (1+1). (1+1) = 8 divisors. So see, that’s the idea isn’t it ? Pretty much of plug and play. We actually get to control how many divisors the number has, once we adjust (n^3). Now you could try some advances…
Okay, so as we just understood we need to achieve a count of 110 divisors. If we have 110.(n^3) = 2^(10). 5^(4). 11 which actually conforms to : (10+1).(4+1).(1+1) = 11.5.2 = 110 So, that implies : (n^3) = 2^(9). 5^(3), which means, n = 2^(3). 5 Now that we have found out n…the rest dosen’t seem really a big deal. You could do it…try !
Well, it’s pretty straightforward now. Let’s call 81.(n^4) equal to some X. First let’s prime factorize 81. That would be 81 = (3^4). So, finally X = (3^4). (2^12). (5^4) How many divisors does that make ? Yes, (4+1).(12+1). (4+1) = 13.25 = 325.
That’s our answer. 325 factors.
Connected Program at Cheenta
Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.