Integers multiple of 11

[Nekkamath]

If a positive three-digit integer that is a multiple of 11 is chosen at random, what is the probability that the sum of its digits is an odd number? Express your answer as a common fraction.

 

[arthur ohlsten]

11x1=11 sum even
11x2=11x11=evwn+even=even
11x3=11x11x11=even plus even + even=sum even
11xn=11+11+... n sums each even =sum even

any multiple of 11 , the sum of the digits is even

Arthur

 

[soroban]

Hello, Nekkamath!

If a positive three-digit integer that is a multiple of 11 is chosen at random,
what is the probability that the sum of its digits is an odd number?

It would seem that all digit-sums of 11n are even, but . . .

. . .\(\displaystyle \begin{Bmatrix}19 \times 11 \:=\:209 \end{Bmatrix}\quad[1]\)
. . \(\displaystyle \begin{Bmatrix}28\times11 \:=\:308 \\ 29\times11\:=\:319 \end{Bmatrix}\quad[2]\)
. . \(\displaystyle \begin{Bmatrix}37\times11 \:=\:407 \\ 38\times 11 \:=\:418 \\ 39 \times 11 \:=\:429 \end{Bmatrix}\quad[3]\)
, , \(\displaystyle \begin{Bmatrix}46\times11 \:=\:506 \\ 47\times11 \:=\:517 \\ 48\times11 \:=\:528 \\ 49\times11 \:=\:539 \end{Bmatrix}\quad[4]\)
. . . . . . . \(\displaystyle \begin{array}{ccc}\vdots & \qquad & \vdots \end{array}\)
. . \(\displaystyle \begin{Bmatrix}82\times11 \:=\:902 \\ 83\times11 \:=\:913 \\ 84\times11 \:=\:928 \\ \quad\vdots\qquad\qquad\vdots \\ 89\times11\:=\:979 \end{Bmatrix}\quad [8]\)


\(\displaystyle \text{It seems there are: }\:1 + 2 + 3 + \cdots + 8 \:=\:36\text{ with }odd\text{ digit-sums.}\)

 

[Deleted member 4993]

If a number "xyz" is divisible by 11 then

x + z - y = 0 --> x + z + y = 2y --> the sum of the digits are always even

or

x + z - y = 11 --> x + z + y = 11 + 2y --> the sum of the digits are always odd and y < 8

The numbers could be:

979.....869......759
..........968......858
.....................957...........etc.

 

[arthur ohlsten]

if you take the sum of the digits of 29 times 11 is it even?

29x11= sum of 29 11's
29x11= sum of 29 even numbers

BUT!!!!!!
29x11= 319
sum of 319=13
sum of 13=4 even

Whether the sum of the digits is even or odd, probably depends on the definition of "the sum of the digits"

I am sorry I defined it in a manner your instructor would not accept. Sorry

Arthur

 

[arthur ohlsten]

If the question was "What is the ratio of 3 digit numbers divisible by 11and are even, to the number of 3 digit nu,bers divisible by 11?"

Three digit numbers divisible by 11, or numbers 100<n<999 modulo 11
11times[10,11,12,...89,90] or [90-10+1]= 81 numbers

Three digit numbers divisible by 11 and even between 100 and 999, or 100<n<99 modulo 22
22 times [5,6,7,... 44,45] = [45-5+1]= 41

answer 41/81

This is probably what your instructor wants, either this or my previous answer of sums of even numbers
Arthur

 

[pka]

This is probably what your instructor wants, either this or

Author, why do you question what the instructor meant?
It is very clearly written. It is not at all ambiguous. It is hardly open to multiple interpretations.

If a positive three-digit integer that is a multiple of 11 is chosen at random, what is the probability that the sum of its digits is an odd number?

Using Soroban’s list the answer is: \(\displaystyle \frac{36}{81}\).

 

[arthur ohlsten]

pka or soroban

how did you get 36. Below I show the derivation for 41! What is in error?
81 as total number of 3 digit numbers divisible by 11 is correc

3 digit numbers divisible by 11 and even would be ," 3 digit numbers divisible by 22"

22n n=[5,6,7,8 ...45]
total 3 digit numbers divisible by 22 = [45-5]+1
3 digit nos. divisible by 2 and 11 = 41

answer 41/81

how did you get 36?

 

[pka]

Here is the list.
209, 308, 319, 407, 418, 429, 506, 517, 528, 539, 605, 616, 627, 638, 649, 704, 715, 726, 737, 748, 759, 803, 814, 825, 836, 847, 858, 869, 902, 913, 924, 935, 946, 957, 968, 979.
I made it using MathCad. If you want the code I will post it.

 

[arthur ohlsten]

why isn't the following on your list ?
11x10=110
11x11=121
11x12=132
11x13=143
11x15= 165
11x16=176
11x17=187
11x18=198
11x19=209 on the list
11x20=220
11x21=231
11x22= 242

There are more
your list is NOT 100 <nos.<999
or 11xn 10<n<90

there should be [90-10]+1=81 terms
40 of these are odd
41 are even

Arthur

 

[arthur ohlsten]

OOPs
I calculated how many terms were even. I did not sum the the digits of the 81 terms.
Accept my apologies
The number 36 is probably correct. I calculated incorrectly twice.
Sorry
Arthur