Question

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 \(\boxed{56}\).