For today, we will discuss the problem given on January 24, 2017:
Problem: How many positive two-digit integers are factors of both $360$ and $540$?
Solution: There are $10$ two-digit integers that are factors of both $360$ and $540$. Note that $\gcd(360,540)=180$. The question is equivalent to determining the number of two-digit factors of $180$. They are $10, 12, 15, 18, 20, 30, 36, 45, 60, 90$.
A possible approach is to list out all factors of $360$ and $540$. By doing this, one is prone to making mistakes since it takes a lot of effort to cycle through all possible factors of both $360$ and $540$. Due to this, students who used this method to solve the problem may have miscounted the number of two-digit factors both $360$ and $540$ share.
By considering the greatest common divisor of both $360$ and $540$, we avoid counting the unnecessary integers that are exclusively factors of $360$ or exclusively factors of $540$.
To determine all of the two-digit factors of $180$, one should develop a systematic approach to make sure that they are not missing any possible two-digit factors. Start with $1$ and ask yourself, "Is $1$ a divisor of $180$?". If so, add $1$ and $180 \div 1 = 180$ to your list and move onto $2$. Repeat this until you reach up to $13$. Then select all two-digit factors and count them!
If you have any approach that you would like to share, please post below! Also, anyone want to give a shot as to why we can stop at $13$ and not consider any factors of $180$ past $14$ onwards? Let us know your thoughts below!