# Greatest common factor, least common multiple.

#### strantor

Joined Oct 3, 2010
6,031
I'm trying to help my neice with her homework over the internet, via her mom who's also trying to help her. I feel like I'm one of those idiots on "are you smarter than a 5th grader" So, here's what I got (this is the email I am sending her mom, with the homework questions in BOLD, and my answer in italics:
Ok, these problems are "greatest common factor" and "least common multiple" problems.
learn about greatest common factor here:
http://www.math.com/school/subject1/lessons/S1U3L2DP.html

and "least common multiple" here:
http://www.math.com/school/subject1/lessons/S1U3L3DP.html
Least common multiple is the smallest number that a set of numbers can be divided by

In order to learn GCF and LCM, you need to first learn how to do "prime factorization"
learn about that here:
http://www.mathsisfun.com/prime-factorization.html

so, for:
"three salesmen make trips at regular intervals, The first every 7 days, the second every
15 days and the third 21 days. If they leave the office on the same day for the first trip,
how many days will elapse before they again leave on the same day?
"
We need to get the least common multiple. we start by finding the prime factors of all 3 numbers:
7: 7
15: 3 X 5
21: 3 X 7
so we have a 3, a 5, and a 7. We multiply them 3X5X7 and get 105. We check our work by seeing if
105 is divisible by 7, 15, and 21. it is.

for:
"Brian is arranging his tanzas in rows. If he puts 11 in a row, there is 1 extra. If he puts 9
in a row, there is non left. What is the smallest number of tanzas that will alow him to do this?
"
I don't understand this question. it says "If he puts 11 in a row, there is 1 extra" which makes me
think there is a total of 12, but then that doesn't make sense with 9 and "none left"

for:
"the product of 2 numbers is 2,700 their greatest common factor is 15 what is their least common multiple"
I'm completely stumped

#### Papabravo

Joined Feb 24, 2006
18,978
I'm trying to help my neice with her homework over the internet, via her mom who's also trying to help her. I feel like I'm one of those idiots on "are you smarter than a 5th grader" So, here's what I got (this is the email I am sending her mom, with the homework questions in BOLD, and my answer in italics:
Brian has 45 tanzas because:
Rich (BB code):
(4 * 11) +1 = 45 = (9*5)

• strantor

#### djsfantasi

Joined Apr 11, 2010
8,444
Strantor said:
for:
"the product of 2 numbers is 2,700 their greatest common factor is 15 what is their least common multiple"
I'm completely stumped

Factor 2,700 into it's prime components:
A*B=3*3*3*2*2*5*5

Shuffling terms around using the knowledge we have that 3*5=15 is the greatest common multiple:
A*B=(3*5)*(3*5)*2*2*3
..and introducing our friends x and y from algebra:
A=15*x
B=15*y

...and using the leftover terms,
x*y=2*2*3

This means there are multiple possibilities for the x and y values.
x= 1, 2, 4
y=12, 6, 3

This results in:
A= 15, 30, 60
B=180, 90, 45

Now using the three possible sets for x and y, I come up with three different answers, depending on which of the three perfectly valid choices we make for the 2 numbers whcih multiply together to get 2,700:
A= 3*5 =15
B= 2*2*3*3*5 =180
Least common multiple =2*2*3*3*5 =180

A= 2*3*5 =30
B= 2*2*3*5*5 =90
Least common multiple =2*2*3*5*5 =90

A= 2*2*3*5 =60
B= 3*3*5 =45
Least common multiple =2*2*3*3*5 =180

Your neice probably needs one correct answer. I like A=30 and B=90 because it stared me right in the face. In fact, I almost convinced myself it was the only answer.

#### strantor

Joined Oct 3, 2010
6,031
@djsfantasi
I thought it was going to be a lot simpler than that. I think this stuff is supposed to be pre-algebra. Thanks for the answer, I will let her know.

#### Zazoo

Joined Jul 27, 2011
114

...and using the leftover terms,
x*y=2*2*3
------------------------------
Now using the three possible sets for x and y, I come up with three different answers, depending on which of the three perfectly valid choices we make for the 2 numbers whcih multiply together to get 2,700:
A= 3*5 =15
B= 2*2*3*3*5 =180
Least common multiple =2*2*3*3*5 =180

A= 2*3*5 =30
B= 2*2*3*5*5 =90
Least common multiple =2*2*3*5*5 =90

A= 2*2*3*5 =60
B= 3*3*5 =45
Least common multiple =2*2*3*3*5 =180

Your neice probably needs one correct answer. I like A=30 and B=90 because it stared me right in the face. In fact, I almost convinced myself it was the only answer.
One correction, of the remaining factors of 2*2*3 the pair of twos must remain in A or B, if you put a single two in both A and B then the GCF becomes 30 not 15. So the second one won't satisfy the required GCF

Last edited:
• djsfantasi

#### Georacer

Joined Nov 25, 2009
5,182
What's a tanza anyway?

I 'd like to justify Papabravo's answer:

We know that:
$$11 \cdot N +1 =9 \cdot K \text{, where }N,K \in \mathbb{N} \Leftrightarrow K=\frac{11 \cdot N +1}{9}$$

The first N that makes K an integer is N=4 and K=5

• strantor