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# help plz

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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?

Jul 6, 2018

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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

Jul 9, 2018