2 teams played each other recently. Team 1 scored N points, while team 2 scored M. 1<M and N<100.
The first digit of N is a, the second digit b. N-a+b^2.
The first digit of M is b, the second is a. By how many points does team M win the game? There is a mathematical way to do this, although I reached the answer by process of elimination.
I'm sorry, N=a+b^2
Let N = 10a + b
but N = a + b^2
so b^2 + a - 10a + b
b^2 - b - 9a = 0
b(b-1) = 9a
which tell me that the product of 2 consecutive numbers, which are less than 10, must be a multiple of 9
only possibilities
9x8 = 72 yes
8x7 = 56 no
7x6 = 42 no
6x5 = 30 no
5x4 = 20 no
4x3 = 12 no
3x2 = 6 no
2x1 = 2 no
so 9a = 72
a = 8
then b^ - b - 72 = 0
(b-9)(b+8) = 0
b = 9
N = a + b^2 = 8 + 81 = 89
M = 98
To determine by how many points Team M (the team with the higher score) wins the game, we need to calculate the difference between their scores.
Let's break down the process step by step:
1. Start with the given information that Team 1 scored N points, and Team 2 scored M points. We know that 1 < M and N < 100.
2. Extract the first and second digits of N. Let's call the first digit "a" and the second digit "b".
For example, if N = 54, then a = 5 and b = 4.
3. Calculate N - a + b^2.
Using the example above, N - a + b^2 would be 54 - 5 + 4^2 = 54 - 5 + 16 = 65.
4. Extract the first and second digits of M. Let's call the first digit "b" and the second digit "a".
For example, if M = 43, then b = 4 and a = 3.
5. Calculate M - b + a^2.
Using the example above, M - b + a^2 would be 43 - 4 + 3^2 = 43 - 4 + 9 = 48.
6. Finally, subtract the result obtained in step 3 from the result obtained in step 5 to find the difference in points.
In our example, the difference would be 48 - 65 = -17, indicating that Team M won the game by 17 points.
It's worth noting that the process used here is a mathematical approach to solving the problem. However, as you mentioned, you reached the answer through a process of elimination. Both methods lead to the correct solution.