# Remembering multiplication of these two numbers: $7 \times 8 = 56$ and $9 \times 6 = 54$

I have almost mastered multiplication table up to 9x9 however, I'm having problems with the following two.

7 x 8 = 56 and 9 x 6 = 54

For some reason my brain thinks that 56 and 54 are somewhat the same and sometimes confuses the two multiplication problems. I was wondering if there is a way I can remember this easier. If someone on here has some rhyming of some sort.

When in doubt I always resort to doing 49+7 or 45+9 to figure out the two, because 7x7 and 9x5 is easy for me.

Thanks

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One possibility: the product of $9$ and $6$ has digits that add up to $9$; the product of $7$ and $8$ does not. – Arturo Magidin Sep 17 '11 at 23:52
For what it may be worth, when I learned the multiplication tables (oh so long ago), I had trouble remembering $8\times 7$ and $8\times 8$; I used to resort to the same type of computation you do above. Lasted for a couple of years, eventually went away with more and more practice. – Arturo Magidin Sep 17 '11 at 23:56
54=60-6$\text{}$ – anon Sep 18 '11 at 0:01
I have trouble with 7x8 is 56. I usually do 6x8 is 48 (which I remember ad 6x8 = 5x8+8=40+8=48) so 7x8 is 48 + 8. Remembering 7x8 =49 +7 works just as well. I never had trouble with 5x9 but I memorized all the nines down cold by knowing that if you add the digits of a multiple of 9 the add to 9. And if you list the multiples of 9 in a column, the first digits go from 0 to 9 and the second digits go from 9 to 0. That means if you have ?x 9, the first digit will be ?-1. And the second digit will be 9 minus that. So the first digit of 6x9 is 6-1 or 5. The second digit is 9-5 = 4. – fleablood May 20 at 6:40
I had the same problem you did (when I was $8$). The solution? I memorized the multiplication table like one memorizes the ABC's. The thing is, one shouldn't have to actually think to know what the answer to $9 \cdot 6$ is. So, if you're having difficulty with it, you simply need to work hard and memorize rather than rely on mnemonics. – MathematicsStudent1122 May 20 at 6:56

## 5 Answers

Multiples of $9$ in the multiplication table have the property that the sum of its digits is always $9$. So $9 \times 6$ cannot be $56$. Also, $9 \times n$ starts with digit $n-1$.

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Getting the 9's table to memory used to be tough for me until I was told of a shortcut; the key is to note the symmetry exhibited by the 9's table:

$$\begin{matrix}09&90\\18&81\\27&72\\36&63\\45&54\end{matrix}$$

where the ones on the left going downward are from $9\times1$ to $9\times5$, and the ones on the right going upward are from $9\times6$ to $9\times10$. (This gives an insight into why.) When I was very young, I had difficulties remembering the multiplication tables; after being told of this property, I figured I could "fold over" the 9's table in half and get away with just remembering the first five, and that part of the multiplication tables subsequently became less difficult for me.

Take heed also of lhf's comments. Any multiple of 9 should have its digits sum to 9 or a multiple of 9. Even if you extend the table to, say, $9\times 13=117$, the digits of the answer should still sum to 9 or a multiple: $1+1+7=9$.

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Multiplying by $9$ is relatively easy, since $9a=10a-a$. I.e., multiply by ten and then subtract. (And multiplication by ten is just adding zero at the end.)

Examples:

$6\times 9 = 6\times 10 - 6 = 60 - 6 = 54$

$12\times 9 = 12\times 10 - 12 = 120 - 12 = 108$

$16\times 9 = 16\times 10 - 16 = 160 - 16 = 144$

$24\times 9 = 24\times 10 - 24 = 240 - 24 = 216$

$88\times9 = 88\times 10 - 88 = 880 - 88 = 792$

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My 4th grade teacher had a story about a guy who was so dumb he was given a test to fill out:

2x9=

3x9=

4x9=

5x9=

6x9=

7x9=

8x9=

9x9=

The guy was utterly dumb and had no idea what to do. So he just wrote the numbers 1,2,3 down the paper until he reached the bottom. The he started from the bottom and wrote the numbers 1,2,3 up the paper until he reached the top. So this is what he wrote:

2x9= 18

3x9= 27

4x9= 36

5x9= 45

6x9= 54

7x9= 63

8x9= 72

9x9= 81

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Try to see it in as many ways as you can (as when you use $49+7$ and $45+9$). For $9$ is nice that (when $n \in (0,1,\dots,9)$) $$9 \times (n+1)= n\cdot 10+(9-n)$$ so $9 \times (5+1)= 5\cdot 10+(9-5)$ (its ugly to see but nice to understand).

A nice exercise could be to write the multiplication table just with the number factorized, I mean $56=7 \times 2^3$ and $54=3^3\times2$.

However, if you're looking for "mnemotechnical stuffs", as to memorize $\pi$'s digits, the best in my opinion is the Phonetical Conversion. I'm surprised is still not being teached at school: with little effort you gain a lifetime powerfull instrument. With that, the multiplication table will be just a picture in your mind.

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