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The LCM of a pair of whole numbers is 450 and the GCF of the numbers is 6. One of the numbers is 18. What is the other number?

 

Let $\frac mn$ be a fraction, where $m$ and $n$ are positive integers. Consider the operation defined by replacing $\frac mn$ by $\frac{m+1}{n+1}$ and then writing the result in lowest terms. For example, applying this operation to $\frac{5}{14}$ would give $\frac{2}{5}.$ How many times must this operation be repeatedly applied to $\frac{1}{2005}$ before we obtain $\frac{2004}{2005}?$

 

How many ordered pairs of positive integers $(m,n)$ satisfy $\text{GCD}(m,n) = 3$ and $\text{LCM}(m,n) = 108?$

 Oct 5, 2019
 #1
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The LCM of a pair of whole numbers is 450 and the GCF of the numbers is 6. One of the numbers is 18. What is the other number?

 

450 / 18 * 6 =150 - the 2nd number.

 Oct 5, 2019
 #2
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Let $\frac mn$ be a fraction, where $m$ and $n$ are positive integers. Consider the operation defined by replacing $\frac mn$ by $\frac{m+1}{n+1}$ and then writing the result in lowest terms. For example, applying this operation to $\frac{5}{14}$ would give $\frac{2}{5}.$ How many times must this operation be repeatedly applied to $\frac{1}{2005}$ before we obtain $\frac{2004}{2005}?$

 

This is what I got (if I understood your question):

1/2005=2/2006=1/1003=2/1004=1/502=2/503=4/504=1/126=2/127=3/128=4/129=5/130=1/26=2/27=3/28=4/29=5/30=1/6 =16 operations not including 1/2005 + 2004 -1 + 1 = 2020 operations. You might want to include  the first one as well{count them!}.

 Oct 6, 2019
 #3
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I think I missed few:

 

2/2006=1/1003=2/1004=1/502=2/503=4/504=1/126=2/127=3/128=4/129=5/130=1/26=2/27=3/28=4/29=5/30=1/6=2/7=3/8=4/9=5/10=1/2 =22 + 2004 =2026 ??. Check my numbers carefully.

 Oct 6, 2019
 #4
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Are you sure 2/503 goes to 4/504?

Guest Oct 6, 2019

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