# Understand the problem

##### Source of the problem

##### Topic

##### Difficulty Level

##### Suggested Book

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**.

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# Connected Program at Cheenta

#### Math Olympiad Program

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.