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I was about to "guess" 2 based on the same premise: first three and a "feeling" about the last two (:
If you say that getting the money
is the most important thing
You will spend your life
completely wasting your time
You will be doing things
you don't like doing
In order to go on living
That is, to go on doing things
you don't like doing
Everyone should know the squares - at least for 1 to 20. After that, you just need to know that multiples of ten are (x * 10)^2 = x^2 * 100. You can then do the halfways [numbers ending in 5] (x * 10 + 5)^2 = ((2x + 1) * 10)^2 / 4 [looks a lot more complicated than it is]; thereafter, for numbers ending in 1, 2, 6, 7 you apply (x + 1)^2 = x^2 + 2x + 1 (or x^2 + (x + 1) + x) [do it twice for 2 and 7] and for numbers ending in 3, 4, 8, 9 you apply (x - 1)^2 = x^2 - 2x -=1 or x^2 - (x + 1) - x = x^2 - x - x - 1 [do it twice for 3 and 8]
At least, that's what I use! And I assure you, once you've got the hang of them that are simple.
Edit: It is also useful to memories powers of two and squares of prime numbers
About 160,000,000 - my mental arithmetic runs out of registers to remember digits in after 5 or 6 digits ... and long division starts using 'em up fast ...
But some of it is easy: 102*112*122*132*142 == 100 * 121 * 144 * 169 * 196
(10*10 is easy, and each square adds 2 more than the previous: 100 -> 121 Adds 21, 121 -> 144 adds 23, so the next two terms are 144 + 25 and 144 + 25 + 27)
We were rote taught our "times tables" up to 12 by 12, so the first three are imprinted on my brain...
long multiplication in your head is reasonably easy as long as you keep the decimal places straight.
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If something has a solution... Why do we have to worry about?. If it has no solution... For what reason do we have to worry about?
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Though one mistake, (a+b)² + (a-b)² = 2a² + 2b². Good we are still as intelligent as then (?).
Probably the same trick the teacher would demonstrate. It would be interesting if some mathematician historian would check whether such tricks were indeed collected for instruction - of numerical math.
1. Each square is approximately 20 more than the previous, so 5x100+20+40+60+80=700, estimating a correction for the approximation and assuming a whole number solution as it's a mental arithmetic problem then the total is 730 and the answer is 2.