+0  
 
0
160
1
avatar+159 

Consider two positive even integers less than 15 (not necessarily distinct). When the sum of these two numbers is added to their product, how many different possible values may result?

xXxTenTacion  Jul 6, 2018
 #1
avatar+20549 
+1

Consider two positive even integers less than 15 (not necessarily distinct).

When the sum of these two numbers is added to their product, how many different possible values may result?

 

\(\text{Let $n_1\lt 15$ is an positive even integer.} \\ \text{Let $n_2\lt 15$ is an positive even integer.} \\ \text{The sum of these two numbers is added to their product $n_1+n_2+n_1\cdot n_2 $.} \)

 

\(\text{Because $n_1$ is even, we can set: $n_1 = 2i \Rightarrow i_{\text{min}} = 1 $ and $i_{\text{max}} = 7 $ } \\ \text{Because $n_2$ is even, we can set: $n_2 = 2j \Rightarrow j_{\text{min}} = 1 $ and $j_{\text{max}} = 7 $ } \)

 

\(\begin{array}{|rcll|} \hline && n_1+n_2+n_1\cdot n_2 \\ &=& 2i + 2j + 2i2j \\ &=& 2(i+j+2ij) \quad & 1 \leq i \leq 7 \qquad 1 \leq j \leq 7 \\ \hline \end{array} \)

 

\(2(i+j+2ij) \quad 1 \leq i \leq 7 \qquad 1 \leq j \leq 7 \\ \begin{array}{|c|r r r r r r r|} \hline (i,j) & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline 1 & 8 & 14 & 20 & 26 & 32 & 38 & 44 \\ 2 & & 24 & 34 & \color{red}44 & 54 & 64 & 74 \\ 3 & & & 48 & 62 & 76 & 90 & 104 \\ 4 & & & & 80 & 98 & 116 & 134 \\ 5 & & & & & 120 & 142 & 164 \\ 6 & & & & & & 168 & 194 \\ 7 & & & & & & & 224 \\ \hline \end{array}\)

 

Only 44 is double.

 

There may result 27 different possible values

 

laugh

heureka  Jul 9, 2018

26 Online Users

avatar
avatar

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.