Can You Find All 3 Digit Numbers Divisible by 11 with a Special Property?

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In summary, the purpose of finding all 3 digit numbers is to identify and list all possible combinations of 3 digits, from 000 to 999. This is useful for various mathematical and statistical analyses, as well as for creating games or puzzles. There are 900 3-digit numbers, ranging from 000 to 999, with the smallest number being 100 and the largest being 999. To find all 3 digit numbers, one can list all possible combinations or use a mathematical formula. In statistics, finding all 3 digit numbers is important for creating random samples, data sets, and probability distributions.
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Determine all 3 digit numbers P which are divisible by 11 and where \(\displaystyle \frac{P}{11}\) is equal to the sum of the squares of the digits of P.
 
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Let $A$, $B$, and $C$ be the 3 digits from left to right, so that:

\(\displaystyle P=100A+10B+C\)

If P is divisible by 11, then we must have:

\(\displaystyle A+C-B=11k\) where \(\displaystyle 0\le k\in\{0,1\}\)

and we require:

\(\displaystyle \frac{P}{11}=A^2+B^2+C^2\)

Case 1: \(\displaystyle k=0\,\therefore\,B=A+C\)

\(\displaystyle 100A+10(A+C)+C=11\left(A^2+(A+C)^2+C^2 \right)\)

\(\displaystyle 11(10A+C)=11\left(2A^2+2AC+2C^2 \right)\)

\(\displaystyle 10A+C=2A^2+2AC+2C^2\)

The only valid solution is:

\(\displaystyle A=5,\,C=0\implies B=5\)

and so the number is 550.

Case 2: \(\displaystyle k=1\,\therefore\,B=A+C-11\)

\(\displaystyle 100A+10(A+C-11)+C=11\left(A^2+(A+C-11)^2+C^2 \right)\)

\(\displaystyle 11(10A+C-10)=11\left(2A^2+2AC-22A+2C^2-22C+121 \right)\)

\(\displaystyle 2A^2+2AC-32A+2C^2-23C+131=0\)

The only valid solution is:

\(\displaystyle A=8,\,C=3\implies B=0\)

and so the number is 803.
 
  • #3
MarkFL said:
Let $A$, $B$, and $C$ be the 3 digits from left to right, so that:

\(\displaystyle P=100A+10B+C\)

If P is divisible by 11, then we must have:

\(\displaystyle A+C-B=11k\) where \(\displaystyle 0\le k\in\{0,1\}\)

and we require:

\(\displaystyle \frac{P}{11}=A^2+B^2+C^2\)

Case 1: \(\displaystyle k=0\,\therefore\,B=A+C\)

\(\displaystyle 100A+10(A+C)+C=11\left(A^2+(A+C)^2+C^2 \right)\)

\(\displaystyle 11(10A+C)=11\left(2A^2+2AC+2C^2 \right)\)

\(\displaystyle 10A+C=2A^2+2AC+2C^2\)

The only valid solution is:

\(\displaystyle A=5,\,C=0\implies B=5\)

and so the number is 550.

Case 2: \(\displaystyle k=1\,\therefore\,B=A+C-11\)

\(\displaystyle 100A+10(A+C-11)+C=11\left(A^2+(A+C-11)^2+C^2 \right)\)

\(\displaystyle 11(10A+C-10)=11\left(2A^2+2AC-22A+2C^2-22C+121 \right)\)

\(\displaystyle 2A^2+2AC-32A+2C^2-23C+131=0\)

The only valid solution is:

\(\displaystyle A=8,\,C=3\implies B=0\)

and so the number is 803.

Well done, MarkFL!(Clapping)
 

FAQ: Can You Find All 3 Digit Numbers Divisible by 11 with a Special Property?

What is the purpose of finding all 3 digit numbers?

The purpose of finding all 3 digit numbers is to identify and list all possible combinations of 3 digits, from 000 to 999. This can be used for various mathematical and statistical analyses, as well as for creating games or puzzles.

How many 3 digit numbers are there?

There are 900 3-digit numbers, ranging from 000 to 999. This is because each digit can take on 10 possible values (0-9), and there are 3 digits in a 3-digit number, making the total number of combinations 10 x 10 x 10 = 1000. However, since 000 is not considered a 3-digit number, the total number of 3-digit numbers is reduced to 1000-1 = 999.

What is the smallest and largest 3 digit number?

The smallest 3 digit number is 100, while the largest is 999. This is because the first digit of a 3-digit number cannot be 0, so the smallest possible number is 100. The largest possible number is 999, as any number higher than that would require a 4th digit.

How do you find all 3 digit numbers?

To find all 3 digit numbers, you can start by listing all possible combinations of the hundreds, tens, and ones digits. For example, you can start with 100, then 101, 102, and so on until 999. Alternatively, you can use a mathematical formula, where the first 3-digit number is 100 and each subsequent number is 1 more than the previous number. For example, 100 + 1 = 101, 101 + 1 = 102, and so on.

What is the importance of finding all 3 digit numbers in statistics?

In statistics, finding all 3 digit numbers can be useful for creating random samples for surveys or experiments. By listing all possible combinations, researchers can ensure that their sample is representative of the entire population. This can also be used for creating data sets for analysis, as well as for creating probability distributions for various variables.

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