Mathematical tricks to check the result of a multiplication

Question

I wondered myself about some tricks allowing quick checks to know whether the result of an operation can be true or is surely false. I found several for square roots on Internet, and so I looked for multiplication, but trying to found them by myself.

Thus, I made some assertions that worked in my tests, but I would like to know if they are valid for every number in, at least, $\Bbb D$. And of course, if they are more tricks hinting toward maybe right or surely wrong. I would like for these hints to be the simplest possible, using only basic operations or "visual" checking and no advanced knowledge in mathematics (but any trick is appreciated).

Let's take three numbers, $a$,$b$ and $c$, with $c = a \cdot b$ ; $a_i$, $b_i$, and $c_i$ are the integer parts of the numbers, and $a_d$, $b_d$, and $c_d$ the decimal part. Putting $\#$ before any of these variable implies we use the number of digits of said variable. As of now, the assertions are the following :

• $\#c_i = \#a_i + \#b_i - 1$
• $\#c_d = \#a_d + \#b_d$
• $\#c = \#a + \#b - 1$

Thanks for the answers, Méta

(Sorry if I mistag the question, I have no mathematical background whatsoever)

Answer 1

Concerning the title question: one method to check the result of a multiplication for the integer parts, say $5\cdot 34=170$ is modular arithmetic. For example, if we compute $5\cdot 34=a$, then a quick check is $$ a\equiv 0 \bmod 2, \; a\equiv 0\bmod 5,\; a\equiv 0 \bmod 17. $$

Answer 2

The classic method of casting out nines is actually just arithmetic modulo $9,$ since this is very easy with our decimal system.

Thus to check $23×14=282,$ for example, note that you only need check $5×5=7\ne3\pmod 9,$ so that the computation is faulty.

Answer 3

$99 \times 99 = 9801$ has four digits, not three, so for the integer parts it should be $\#a_i + \#b_i - 1 \leq \#c_i \leq \#a_i + \#b_i$ for all $a, b$ with $|a|, |b|\geq 1$. This can be verified by using the fact that $\#x_i = 1 + \lfloor\log_{10}|x|\rfloor$ for $|x| \geq 1$.

For $|a| < 1$ or $|b| < 1$ this is no longer the case: for example $100000 \times 0.001 = 100$. In the general case, you can only say $\#x_i = 1+\max(0,\lfloor\log_{10}|x|\rfloor)$ so the only general relation remaining is $\#c_i \leq \#a_i + \#b_i$.

For the decimal parts, again it's not that easy: For example $0.125 \times 0.08 = 0.01$, or $0.01 \cdot 10 = 0.1$. A correct inequality would be $\#c_d \leq \#a_d + \#b_d$. This can be verified using the $2$-adic and $5$-adic valuations of the numbers involved.

Answer 4

Take the sqrt of the product. then divide by the lower argument of the multiplication. Divide the higher by the result. The result of this last division, should give you back your sqrt.