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Number Theory I-B

 
 
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Number Theory I-B
by Amber Lin - Thursday, August 22, 2019, 4:16 PM
 

In Chapter 10, I have the following questions:

Question 7- Why do you use specifically even powers to find perfect squares? In the answer key, it states to do 3 times 3= 9 for the final answer. Why?


Question 9- In the answer key it states a=8, and b=8. However, shouldn't a and b be separate and different integers? Therefore shouldn't the solution be 7842, instead of 7848?

 
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Re: Number Theory I-B
by David Reynoso - Friday, August 23, 2019, 10:48 AM
 

Question 7: A number $n$ is a perfect square if $\sqrt{n}$ is an integer. If we factor out $n$ as a product of primes $n = p_1^{e_1}\cdot p_2^{e_2}\cdots p_k^{e_k}$ for $n$ to have an integer square root we would need to be able to split all the factors in two equal groups, thus we need for each of the exponents $e_i$ to be even: $\sqrt{n} = p_1^{e_1/2}\cdot p_2^{e_2/2}\cdots p_k^{e_k/2}$. In this particular problem we want to find perfect squares that divide $2593 = 2^5 \cdot 3^4$. Any factor of this number will be of the form $2^a \times 3^b$ with $a = 0,1,2,3,4,5$ and $b = 0,1,2,3,4$. Since we want the factors to be perfect squares, we need that $a$ and $b$ are both even, so that reduces our options of $a$ and $b$ to $a = 0, 2, 4$ and $b = 0, 2, 4$. Thus, we have three options to choose from for the exponent for $2$ and three options to choose from for the exponent of $3$, so there are $3 \cdot 3 = 9$ distinct perfect squares that divide $2592$.

Question 9: The problem is asking for the largest possible value of $\overline{7a4b}$, but it doesn't say that all digits are distinct, so it is possible that $a$ and $b$ have the same value.

Picture of Amber Lin
Re: Number Theory I-B
by Amber Lin - Saturday, August 24, 2019, 7:31 PM
 

For Lesson 7, what is the meaning of the second sentence's formula?

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Re: Number Theory I-B
by David Reynoso - Tuesday, August 27, 2019, 10:59 AM
 

The expression $n = p_1^{e_1}\cdot p_2^{e_2}\cdots p_k^{e_k}$ represents the prime factorization of the number $n$. The numbers $p_i$ are distinct prime numbers, and the numbers $e_i$ are the exponents for each of the prime numbers in the factorization. 

  • Say $n = 72$, so its prime factorization is  $n = 2^3 \times 3^2$. Since there are three $2$'s and two $3$'s in the prime factorization of $72$, it is not possible to split them in two equal groups, the closest would be $(2^2 \times 3) \times (2 \times 3)$, so $72$ is not a perfect square.
  • Say $n = 36$, so its prime factorization is $n = 2^2 \times 3^2$. This time it is possible to do $n = (2 \times 3) \times (2 \times 3)$, since we have an even number of each factor. So $\sqrt{n} = 2 \times 3$. (Note both $n$ and $\sqrt{n}$ have the same prime factors, and exponents of the prime factorization of $\sqrt{n}$ are exactly half of the exponents of $n$.)