# Proof behind this mental math multiplication?

Recently came across this technique of multiplying two $2$-digit numbers involving the same digit at the tens place and the sum of digits at units place being $10$. E.g., $73 X 77$ has same digit at the tenth place, viz., $7$ and the sum of digits at the ones place is $7+3 = 10$.

As per the technique the easy way to solve this simply multiply the digits at ones place together which in the above case would be $7. 3 = 21$ and the digit at the tenth place by the next number, i.e., $7.8=56$ in the above. The final answer then becomes after concatenating the two answers thus obtained, and hence $73.77=5621$. Similarly, $68.62=4216$ obtained by multiplying $6.7=42$ and $8.2=16$ and concatenating the two to obtain $4216$.

Can anyone explain the math behind this seemingly simple math trick?

• (70 + 3)(70 + 7) = 70*70 + 70*3 + 70*7 + 3*7 = 70*80 + 3*7 = 7*8*100 + 3*7 – mdave16 Oct 5 '17 at 10:53
• it's fairly easy to know if you know FOIL. – user451844 Oct 5 '17 at 10:53
• Thanks. I hate this about mental math trick books...they never explain the math behind things and just give the method. – naveen dankal Oct 5 '17 at 10:54

$(10x + a)(10x +b) = 10x(10x+a+b) + ab$. Furthermore, if $a+b = 10$ , then this simplifies to $10x(10x + 10) + ab = 100x(x+1)+ab$.

That is, what we are doing, is basically the following: given say $72 \times 78$, we get $x(x+1) = 56$, and $ab = 16$, so the answer is $5616$. The reason why you can just join $56$ and $16$ is because $ab$ is always a two digit number, and $100x(x+1)$ has it's last two digits as $00$, so linear addition becomes concatenation.

By the way, this assumes nothing about $x$ at all. For example: $$173 \times 177 = (17 \times 18) *(3 \times 7) = (306)*(21) = 30621$$

So $x$ could be anything. All we require is that $a+b = 10$ and all but the last digit of the numbers are the same.

A special case arises when $a=b=5$ for then $ab=25$, and essentially you are squaring the number $(x5)^2$. For example, $75^2 = (7 \times 8)*25 = 5625$.

• I edited because in 173X177 it should have been 3X7 instead of 3X9 – naveen dankal Oct 5 '17 at 11:05
• Yes, that is correct. thank you for the edit. By the way, I happen to know many of these techniques. The one you are learning can be generalized to the base method of multiplication, as it is called. – астон вілла олоф мэллбэрг Oct 5 '17 at 11:07
• Can you suggest me a good book that explains the techniques too? – naveen dankal Oct 5 '17 at 11:07
• I never read from sources. I always either looked up material online, or was told by somebody. In this case, I was told by a teacher. In the end, though, everything has an algebraic interpretation. For example, from $(x-a)^2 = x^2 -2ax + a^2$, you can make your own rule like this : for $90 \leq y \leq 100$, the square of $y$ is this : find $z = 100 - y$, then form the number $(100 - 2z) * z^2$. For example, $y = 93$ then $z = 7$ and the square is $(100-14) * 7^2 = 8649$, which is correct. So I urge you to experiment, it will bring rich rewards. – астон вілла олоф мэллбэрг Oct 5 '17 at 11:13
• Thanks for the suggestion – naveen dankal Oct 5 '17 at 11:17