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Get out
your paper & pencil, because I have a new puzzle for you!!
Responses: PUZZLE #107 More about nines (revised)
We
saw some properties of nines in Puzzle 106, here are some more.*
A. As
in #106, take any 2 digit number ab, a and b not equal, reverse the digits, and
subtract. The result is always a multiple of 9.
Examples:
43-34 = 9,
83-38
= 45 = 5x9,
79-97
= -18 = -2x9.
Explain
why this works.
Actually
you can use any size number of digits (why?).
This is like Puzzle
106 which was for larger numbers.
Let
ab be a 2 digit number with a & b different.
Then
if ba is larger than ab we have
ba
– ab = (10b + a) – (10a + b) = 10b – b – 10a + a = 9b
– 9a = 9(b-a).
You
may have wondered why we didnt have a Mystery Number for 2 digit numbers?
It
turns out that these numbers do not reduce to a single number when you subtract
them from their reverse: ab – ba is a multiple of 9, but it will
determine another multiple of 9, and so on in a circle.
Example:
72 - 27 = 45, 54 – 45 = 9, 90 – 09 = 81, 81 – 18 = 63, 63
– 36 = 27, 72 – 27 = 45, . . .
B. Take
any 2 digit number ab, add the digits together, a+b, and subtract the result
from ab, ab-(a+b). The result is always a multiple of 9.
Examples:
13-(1+3) = 13-4 = 9,
43-(4+3)
= 43-7= 36 = 6x9.
Can
you explain why this works?
Using an argument
similar to Part A., let ab be the 2 digit number, then
ab
– (a+b) = (10a + b) – (a+b) = 10a – a + b – b = 9a.
C. Given
any multiple of 9, the sum of the digits of the answer will always be a
multiple of 9.
Examples:
8x9 = 72, and 7+2 = 9,
121x9
= 1089, and 1+8+9 = 18 = 2x9.
Why
does this work?
The key here is to
notice that a multiple of 9 can always be written as a sum m of 10s and 9-m, so
that the number m of 10s plus 9-m will equal 9 or a multiple of 9. For example
2x9 = 18 = 10 + (9-1), and 1 + (9-1) = 9.
*** Another
way to look at this is in terms of the result in Part B. There we saw that the
equation ab – (a+b) = 9a
holds. Now if ab is also a multiple of 9, say ab = 9n, then the equation would
look like this: 9n – (a+b) = 9a. So if we solve for (a+b) we have (a+b) =
9n – 9a = 9(n-a), and (a+b) is a multiple of 9.
D. The sum
of the digits of a number is called the digital root of the number. The
number can be any size. By Part C a number is a multiple of 9 if and only if
the digital root is also a multiple of 9.
Explain
why this is true.
By Part C we see that
a multiple of 9 will satisfy the result that the digital root is also a
multiple of 9.
Now
reverse the argument, and assume that the digital root is a multiple of 9. Then
the number m of 10s and the number (9-m) will determine a multiple of 9.
*** Using
the alternate argument in Part C, we have the digital root equal to a multiple
of 9, and conversely, if it is a multiple of 9, say (a+b) = 9m, then by Part B
we have ab – (a+b) = ab – 9m = 9a, so ab = 9a + 9m = 9(a+m), and ab
is a multiple of 9.
E. Casting
out nines: This is a check (some call it a sanity check) on different
arithmetic operations. It is not used so much with hand calculators these days.
Problem:
add up the column: 3264
8415
2946
3206
First
compute the excess of each number (divide the digital root of each number by 9
and list the remainder – this is the casting out operation), then add up
the numbers and all of these excesses:
3264
-> (3+2+6+4) = 9+6 -> 6
8415
-> 0
2946
-> 3
3206
-> 2
17831 -> 11
Now
reduce the answer and the sum of the excesses by 9:
17831
-> (1+7+8+3+1) = 20 -> 2, and 11 -> 2.
This
tells us that the answer 17831 may be correct, in other words in order for the
summation to be correct the two reduced sums must be equal. If the answers do
not agree we know the answer is wrong!
The
technique also works for subtraction, multiplication, and division problems.**
Explain
why it works? (see **)
* See Wikipedia for more information on this:
http://en.wikipedia.org/wiki/9_(number)
** http://en.wikipedia.org/wiki/Casting_out_nines
Have fun!
Send
your comments, ideas and solutions before Monday to the email below, and
be sure to put * MM * in the
subject line: parksjm@potsdam.edu
Visit
us here online at: http://www2.potsdam.edu/parksjm/MM1.1.htm to see the results every Friday/Monday.
See you next
time on MATH MINUTE! (theme music
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