conjunctures

Hello, n_duncan!

Did you even try what they suggested ??

Quote:

Use a calculator to explore the pattern write a conjecture based on what you observe,

$\displaystyle 10)\;\;\begin{array}{c}101 \times 25\:=\:{\color{blue}2525}\\
101 \times 34 \:=\: {\color{blue}3434} \\
101 \times 49\:=\:{\color{blue}4949} \end{array}$

Just a wild guess . . .

But it looks like 101 times any two-digit number $\displaystyle ab$ is equal to $\displaystyle abab$.

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$\displaystyle 11)\;\;\begin{array}{ccc}11 \times 11 &= & {\color{red}121} \\
111 \times 111 & =& {\color{red}12321} \\
1111 \times 1111&=&{\color{red}1234321}\end{array}$

I bet that: .$\displaystyle 11111 \times 11111 \:=\:123454321$

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$\displaystyle 12)\;\;\begin{array}{ccc}3 \times 4 &= & {\color{green}12}\\
33 \times 34 & =&{\color{green}1122}\\
333 \times 334 & = & {\color{green}111222}\end{array}$

And I bet that: .$\displaystyle 3333 \times 3334 \:=\:11112222$