The sum of the digits of a two digit number is multiplied by 8. The result is 8 more than the 2-digit number. Find the
number
If the digits are a and b,
8(a+b) = 8+10a+b
8a+8b = 8+10a+b
7b - 2a = 8
The number is 32
from the above
7b - 2a = 8
2a = 7b-8
a = (7b-8)/2
but a and b can only be a numbers between 0 and 9
and since 7b-8 is divided by 2, the expression
7b-8 must be even.
This means that b must be an even number
let's try values of b
let b=2
a = (14-8)/2 = 3
giving us 32
let b=4
a = (28-8)/2 = 10 , but that is too large, and will only get larger if we pick higher values of b
so all done, the number is 32
Well, well, well, looks like we've got a brain teaser here. Let's clown around with some math and solve this puzzle!
Let's call the tens digit of the two-digit number "x" and the units digit "y". According to the problem, we know that:
x + y = 8(x + y) + 8
Now, let's simplify this equation by distributing the 8:
x + y = 8x + 8y + 8
Hmm, things are getting interesting! Now, let's gather the variables on one side:
x + y - 8x - 8y = 8
Combining like terms:
-7x - 7y = 8
Now, let's simplify some more by dividing the whole equation by -7:
x + y = -8/7
Uh-oh! Something went wrong on this math roller coaster. There's no specific value for the two-digit number. It seems like we've reached a dead end on this clownish adventure. There's no unique solution for this problem.
Well, keep in mind that math doesn't always have a straightforward answer. Sometimes, it can be a bit of a circus act!
To find the number, we can start by breaking down the problem into smaller steps and understanding the given information.
Step 1: Understand the given information:
The sum of the digits of a two-digit number is multiplied by 8, and the result is 8 more than the original number.
Step 2: Represent the number algebraically:
Let's represent the given two-digit number as "10a + b," where "a" represents the tens digit, and "b" represents the ones digit.
Step 3: Write the equation:
According to the given information, the sum of the digits (a + b) is multiplied by 8 and should be equal to 8 more than the original number (10a + b + 8). So, we can write the equation as:
8(a + b) = 10a + b + 8
Step 3: Solve the equation:
Start by simplifying the equation:
8a + 8b = 10a + b + 8
Combine like terms:
8a - 10a + 8b - b = 8
-2a + 7b = 8
Since we are trying to find the values of "a" and "b," we need another equation to solve the system of equations.
Step 4: Use the constraints of a two-digit number:
We know that the first digit cannot be zero since it is a two-digit number. Therefore, "a" should be greater than zero (a > 0). We also know that both "a" and "b" are integers between 0 and 9.
Step 5: Trial and error method:
To find the values of "a" and "b," we can use the trial and error method by substituting different values in the equation -2a + 7b = 8 and checking if they satisfy the conditions.
By trying out different pairs of integers (a, b), we can find the solution that satisfies the equation and the conditions. Let's try substituting values:
Substituting a = 1 and b = 2:
-2(1) + 7(2) = -2 + 14 = 12 ≠ 8
Substituting a = 1 and b = 3:
-2(1) + 7(3) = -2 + 21 = 19 ≠ 8
...
By trying different combinations, we find that there is no integer solution that satisfies the equation. Therefore, there is no two-digit number that meets the given conditions.