The difference between the digits of a two digit number is 1. The number itself is 1 more than five times the Sim of its digit . if the units digit is greater than the tens digit find the number.

Answer 1

Let's denote the two-digit number as \(10t + u\), where \(t\) is the tens digit and \(u\) is the units digit.

From the problem, we have the following information:

1. The difference between the digits is 1: \[ u - t = 1 \]

2. The number itself is 1 more than five times the sum of its digits: \[ 10t + u = 5(t + u) + 1 \]

3. The units digit is greater than the tens digit: \[ u > t \]

Step 1: Solve for \(u\) in terms of \(t\)

From the first equation, we can express \(u\) in terms of \(t\): \[ u = t + 1 \]

Step 2: Substitute \(u\) into the second equation

Now, substitute \(u = t + 1\) into the second equation: \[ 10t + (t + 1) = 5(t + (t + 1)) + 1 \]

Simplifying both sides: \[ 10t + t + 1 = 5(2t + 1) + 1 \]

This simplifies to: \[ 11t + 1 = 10t + 5 + 1 \] \[ 11t + 1 = 10t + 6 \]

Step 3: Solve for \(t\)

Now, isolate \(t\): \[ 11t - 10t = 6 - 1 \] \[ t = 5 \]

Step 4: Find \(u\)

Now, substitute \(t = 5\) back to find \(u\): \[ u = t + 1 = 5 + 1 = 6 \]

Step 5: Form the two-digit number

The two-digit number is: \[ 10t + u = 10(5) + 6 = 50 + 6 = 56 \]

Step 6: Verification

1. The difference between the digits: \[ u - t = 6 - 5 = 1 \quad \text{(correct)} \]

2. The number compared to five times the sum of its digits: \[ 10t + u = 56, \quad t + u = 5 + 6 = 11 \] \[ 5(t + u) + 1 = 5(11) + 1 = 55 + 1 = 56 \quad \text{(correct)} \]

3. Units digit greater than tens digit: \[ u = 6 > t = 5 \quad \text{(correct)} \]

Thus, the two-digit number is \(\boxed{56}\).