1. ## Mental calculation

Find the digit represented by ? in the following problem: 37? × 35 = 13090. No calculators, please.

2. Originally Posted by Sigyn3
Find the digit is represented by ? in the following problem: 37? × 35 = 13090. No calculators, please.
37? * 7 * 5 = 13090

37? * 7 = twice 13090/10 = twice 1309 = 2618 (easy w/o calculator)

only 4 * 7 possible: 2[8], 261[8]

So red? = 4

3. Thanks Wilmer. Your method is much better than the one I had in mind!

4. Hello, Sigyn3!

Find the digit is represented by $\displaystyle {\color{red}?}$ in the following problem: .$\displaystyle 37{\color{red}?} \times 35 \:=\: 13,\!090.$
I did it "mentally", but then my brain is cluttered with various trivia . . .

I saw that: .$\displaystyle 37 \times 35 \:=\:(36+1)(36-1) \:=\:36^1 - 1^2 \:=\:1296 - 1 \:=\:1295$

. . So that: .$\displaystyle 370 \times 35 \:=\:12,\!950$ . . . which is 140 short of 13,090

. . Hence, we need four more 35's.

Therefore: , $\displaystyle 37{\color{red}4} \times 35 \:=\:13,\!090$