Determine all pairs (m, n) of a two-digit natural number m and a single-digit natural number n satisfying the following conditions.
1) If the number n is given as the number between the two digits of the number m, then a three-digit number is obtained which is 11 times the number m.
2) If the number n is used as a number before the decimal number of the number m, then a three-digit number is obtained that is 21 times as large as the number n.
If I understand this correctly, then if the digits of m are a and b, we have
m = 10a+b
and so
100a+10n+b = 11(10a+b)
100n+10a+b = 21n
But this comes out with negative solutions. Maybe you can fix my logic and take it where it's supposed to go.
To determine all pairs (m, n) that satisfy the given conditions, we can set up a system of equations based on the conditions provided.
Let's start by translating the given information into mathematical equations:
Condition 1: If n is given as the number between the two digits of m, a three-digit number is obtained that is 11 times m.
Let's express this condition as an equation:
100 * m + 10 * n + m = 11 * m
Simplifying this equation, we have:
110 * m + 10 * n = 11 * m
Subtracting 11 * m from both sides, we get:
99 * m + 10 * n = 0
Condition 2: If n is used as a number before the decimal point of m, a three-digit number is obtained that is 21 times n.
Let's express this condition as an equation:
10 * n + m = 21 * n
Simplifying this equation, we have:
m = 21 * n - 10 * n
Combining like terms, we get:
m = 11 * n
Now, we have a system of equations:
1) 99 * m + 10 * n = 0
2) m = 11 * n
To find all pairs (m, n) that satisfy these equations, we can substitute the value of m from equation 2 into equation 1.
Substituting m = 11 * n into equation 1, we get:
99 * (11 * n) + 10 * n = 0
Expanding the equation, we have:
1089 * n + 10 * n = 0
Combining like terms, we get:
1099 * n = 0
Dividing both sides by 1099, we have:
n = 0
Since we are looking for single-digit natural numbers for n, the only solution is n = 0.
Now, substituting n = 0 into equation 2, we find:
m = 11 * 0
m = 0
Therefore, the only pair (m, n) that satisfies the given conditions is (0, 0).
In summary, the only pair satisfying the conditions is (0, 0).