A two-digit number equals \(8\) times the sum of its digits minus \(5\), and also equals \(16\) times the difference of its digits plus \(3\). Find the number.
The number is 83.
Step 1: Represent the digits
Let the tens digit be \(a\) and the units digit be \(b\).
So the number can be written as \(10a + b\).
Step 2: Form the first equation
The question says: "The number equals 8 times the sum of its digits minus 5".
Sum of digits = \(a + b\).
So: \(10a + b = 8(a + b) - 5\).
Simplify:
\(10a + b = 8a + 8b - 5\)
Bring all terms to one side:
\(10a - 8a + b - 8b = -5\)
\(2a - 7b = -5\) → Equation (1)
Step 3: Form the second equation
The question also says: "The number equals 16 times the difference of its digits plus 3".
Difference of digits = \(a - b\).
So: \(10a + b = 16(a - b) + 3\).
Simplify:
\(10a + b = 16a - 16b + 3\)
Bring all terms to one side:
\(10a - 16a + b + 16b = 3\)
\(-6a + 17b = 3\) → Equation (2)
Step 4: Solve the two equations
Equation (1): \(2a - 7b = -5\)
Equation (2): \(-6a + 17b = 3\)
Multiply Equation (1) by 3 to eliminate \(a\):
\(6a - 21b = -15\)
Add this to Equation (2):
\((-6a + 17b) + (6a - 21b) = 3 + (-15)\)
\(-4b = -12\)
So, \(b = 3\).
Step 5: Find \(a\)
Put \(b = 3\) into Equation (1):
\(2a - 7(3) = -5\)
\(2a - 21 = -5\)
\(2a = 16\)
\(a = 8\).
Step 6: Write the number
The tens digit is 8, and the units digit is 3.
So the number is 83.