Online Course Discussion Forum

Number Theory I-B

 
 
Picture of Amber Lin
Number Theory I-B
by Amber Lin - Thursday, 15 August 2019, 10:04 PM
 

I have questions on the following problems in Chapter 7:

7.22

7.27

7.28

7.30.

 
Picture of David Reynoso
Re: Number Theory I-B
by David Reynoso - Friday, 16 August 2019, 12:15 PM
 

Hi Amber! Do you have specific questions about this problems? Is there something you have tried so far but got stuck?

Here are some hints:

7.22: On example 7.2 we could see that for any two numbers $m$ and $n$ it is true that $\gcd(m,n) \times \text{lcm}(m,n) = m\times n$. This problem is exploring if the same happens for any three numbers $l$, $m$, and $n$. Try with different values of $l$, $m$ and $n$ to see if it is true or not.

7.27: Remember a number $n$ is a multiple of both $a$ and $b$ if it is a multiple of $\text{lcm}(a,b)$.

7.28: Look into the prime factorizations of $5$, $510$ and $51$ to find clues of what should be the prime factors in the prime factorizations of $A$, $B$ and $C$. Remember the GCD of two numbers is equal to the product of all the common prime factors of with the smallest exponent you can see in their prime factorizations, and the LCM of two numbers is equal to the product of all distinct prime factors you can see in their prime factorizations with the largest exponent you can see.

7.30: Recall a number $n = p_{1}^{e_1} \cdot p_{2}^{e_2} \cdot \cdots \cdot p_{k}^{e_k}$ has exactly $(e_1 + 1)(e_2 + 1)\cdots(e_k + 1)$ factors. For example, If we want a number to have exactly $25$ factors, since $25 = (24 + 1) = (4 + 1) \times (4 + 1)$, the number should be of the form $p^{24}$ for some prime number $p$, or $p^{4} q^{4}$ for distinct prime numbers $p$ and $q$. What would be the case for numbers that have $9$ and $4$ factors?

Picture of Amber Lin
Re: Number Theory I-B
by Amber Lin - Friday, 16 August 2019, 11:35 PM
 
I still am confused about 7.27 and 7.30:


7.27- How would this hint help me? The question states: "Suppose an uncle distributes $1$1 bills among his nieces and nephews. If he distributed the dollars evenly among the nieces, each niece would get 2424 dollars, and if he distributed them evenly among the nephews, each nephew would get 4040 dollars. In fact, he distributes the dollars evenly among all his nieces and nephews. How many dollars does each niece or nephew receive?"  

How would the niece and nephew get the same amount of money when distributed together, while getting different amounts when apart? I understand there may be different amounts of nieces and nephews, but how would I solve that?

7.30- I checked the answer explanation on the test for 7.30, and the first part was "AA must be of the form p8p8 or p2q2p2⋅q2 for primes p,qp,q." How do they get A as the form of p^8?  

Picture of David Reynoso
Re: Number Theory I-B
by David Reynoso - Monday, 19 August 2019, 2:31 PM
 

For 7.27: Since he can evenly distribute the bills among his nieces, each getting $24$ dollars, the number of bills must be a multiple of $24$, and since he can evenly distribute the bills among his nephews, each getting $40$ dollars, the number of bills must also be a multiple of $40$. Thus, the number of bills must be a multiple of both $24$ and $40$. Any number that is a multiple of both, must be a multiple of their LCM, $120$. Say the number of bills was then $120k$ (for some integer $k$), so how many nieces and how many nephews are there? How many nieces and nephews does he have in total? (The answer to this question will be in terms f $k$). Use this, and the fact that you have $120k$ bills to find out how many bills does each of them get when he distributes them among all of them.

For 7.30: The number $A$ has exactly $9$ divisors. Since we can write $9 = (8+1) = (2 + 1)(2 + 1)$, the number must either be $A = p^8$ for some prime number $p$ (since $9 = 8 + 1$), or of the form $A = p^2 \cdot q^2$ for distinct primes $p$ and $q$ (since $9 = (2 + 1)(2 + 1)$).