You expect me to believe it is a mere coincidence that two digit products of multiples of 9 add up to 9?

Believe what you want. it’s a fact.

Think about it – 9 is one less than 10, so anything multiplied by 9 will be (10 times that number) minus (1 times that number).

it’s really pretty simple logic.

@elbanditoroso how does that legitimate that the product’s digits add to be nine??

Sigh.

There are only so many numbers involved here. Just go through them an look at what is happening. Compare:

1×10 = 10
2×10 = 20
3×10 = 30
4×10 = 40
...
9×10 = 90

Is it an “unlikely coincidence” that those all end in 0?

Ok, so now look at:

1×9 = 9
2×9 = 18
3×9 = 27
4×9 = 36
...
9×9 = 81

Compare the two sequences? See what the difference is? It’s one less than the series above, so the number is ALMOST multiplying by 10. It’s not enough less to make the first digit more than one less than 10, 20, 30, and it’s one less that a round multiple of 10 for each 9 you’re adding, so the 1’s digit goes now by 1 while the 10’s digit goes up by 1, so their SUM stays the same, because x + 1 – 1 = x.

Of course, that pattern breaks down at the point the result reaches three digits.

The sum of the digits of any number divisible by 9 is divisible by 9. For two digit numbers the only possible value for the sum is 9.

It gets better. The process of casting out 9’s used to be taught in elementary school to check arithmetic. It works like this. Suppose you want to check that 73+32=105. Add the digits of 7 and 3 to get 10. Then add the 1 and 0 to get 1. Add the digits of 32 to get 5. 1+5=6, which is what you get when you add the digits of 105. It is even more impressive for multiplication.

Here is a feat of magic you can perform. Ask someone to write down a five digit number. Then tell them to form a second number by scrambling the digits of the first number. Then have the person use their calculator to subtract the smaller from the larger. Finally, have them add the digits as explained above. You then tell them what number it is. Do you see what it must be?

I did not indicate how casting out nines got its name. When adding the digits of 73 to get 10, you can also say to subtract 9 to get 10 – 9 =1. It works the same as adding the digits.

@LostInParadise ok, that does it. Nines are witchcraft.

This is all because decimal numbers are base 10 and 9 is one less than 10. If we worked with base 8 numbers, 7 would work the same as 9 does for decimals.

It is a shame that modular arithmetic is not taught in k-12 schools. It may not be very useful for most people, but it is not that difficult to understand and it neatly explains some things like the properties of 9.

Math is kinda witchcraft, period! Kinda magic.

@LostInParadise, Oh, I just finished explaining casting out nines over here. I hope we agree. I am definitely not a math person, but I did retain some things.

I also corrected myself at post 189 because I don’t think I should have included division.

There’s this curiosity too, the reversals:

1×9 = 09    10×9 = 90
2×9 = 18    9×9 = 81
3×9 = 27    8×9 = 72
4×9 = 36    7×9 = 63
5×9 = 45    6×9 = 54

What does it all mean?

As I said previously, all the patterns are because our number system is base 10 and 9 is one less than 10. In base 8, the multiples of 7 start out as 07, 16, 25, 34, 43, 52, 61 and 70. In this case the left hand column stands for 8 instead of 10.

@LostInParadise I can understand the necessity of numbers having these patterns.

I also find it uncanny that they do

Modular arithmetic is not that hard to explain, but I would need a bit of space to give a good explanation. Let me give you a taste for how it works.

Consider 533×726. What is the last digit of the product. That is easy. Just multiply the last digits of the two numbers. 6×3 = 18 so the last digit is 8.

The last digit of a number is the remainder when it is divided by 10. The term for the remainder of a number x after dividing by m, is x mod m. It turns out that for all whole numbers, not just 10, values mod m are multiplicative (as shown previously) and also additive.

Numbers mod 9 are easy to work with because 10 mod 9 = 1. That means that 100 mod 9 = (10 mod 9)x(10 mod 9) = 1. 300 mod 9 = 3x(100 mod 9)=3. So (543 mod 9) = (5+4+3) mod 9 = 3. In general, the remainder of a number when divided by 9 is the same as the remainder of the sum of the digits when divided by 9. In particular, a number is divisible by 9, has no remainder, if the sum of its digits is divisible by 9. When you use the process of factoring out 9’s all you are doing is performing arithmetic mod 9.

@LostInParadise,
> As I said previously, all the patterns are because our number system is base 10 and 9 is one less than 10.

I understand that you said that’s the reason, but I don’t see why that’s the reason. What is the logical link? Why does that explain the patterns of nines, when 2-less-than-10, say, has no such pattern? Sure, it has its own pattern, but it’s nothing like the reversibility, the digits adding up, etc. It’s sort of like my saying, “That comma? It’s because it’s nonrestrictive.” Unless I explain the connection between the comma and the restrictive clause, saying what makes it restrictive, it’s not enlightening.

@Jeruba , My brief explanation of modular arithmetic is the best that I can do. When people do a division, they tend to disregard the remainder, but these remainders have a life of their own, and they are at the heart of modular arithmetic. Put as succinctly as I can, the remainder when divided by n of the sum of two numbers is the sum of their remainders when divided by n, and similarly the remainder of the multiplication of two numbers is the product of their remainders. If we call r(x,n) the remainder of x when dividing by n then r(x+y,n)=r(x,n)+r(y,n) and r(xy,n) = r(x,n)r(y,n)

What is special about 9 is that the remainder of 10 when divided by 9 is 1. Using the multiplication rule for remainders, that means that any number 10 raised to the power of n (n multiplications of 10) will have a remainder of 1 raised to the power of n, which equals 1. For a number like 700, the remainder when dividing by 9 is 7 times the remainder of 100 when divided by 9, giving 7×1 = 7. It follows that to find the remainder when dividing by 9 for a number like 251, you just add up the digits to get 2+5+1=8 as the remainder when divided by 9. If we want to find the remainder of 251×476, we add the digits for each number and then the multiplication rule says that we just multiply those digit sums to get the remainder of 251×476, without having to carry out the full calculation. If someone calculates 251×476, the sum of the digits of the product must match the previously calculated remainder. That is how casting out 9’s works.