The products of a two-digit number and a number consisting of the same digits written in the reverse order is equal to 1612. Find the great number.

Options

A. 84
B. 76
C. 62
D. 43

Answer

To solve the problem, we need to find a two-digit number and its reverse such that their product equals 1612. Let's break down the steps:

### Step 1: Understand the Problem
We need to find two numbers:
- A two-digit number: let's denote it as \( ab \) (where \( a \) is the tens digit and \( b \) is the units digit).
- Its reverse: \( ba \).

The product of these two numbers is given as:
\[ (ab) \times (ba) = 1612 \]

### Step 2: Express the Numbers Mathematically
The two-digit number \( ab \) can be expressed as:
\[ 10a + b \]
And the reverse \( ba \) can be expressed as:
\[ 10b + a \]

### Step 3: Set Up the Equation
We can set up the equation based on the product:
\[ (10a + b)(10b + a) = 1612 \]

### Step 4: Factor 1612
To find suitable two-digit numbers, we can factor 1612. Let's find the factors of 1612:
1. Start by dividing by 2:
\[ 1612 \div 2 = 806 \]
2. Divide 806 by 2:
\[ 806 \div 2 = 403 \]
3. Next, divide 403 by 13:
\[ 403 \div 13 = 31 \]
4. Thus, the prime factorization of 1612 is:
\[ 1612 = 2^2 \times 13 \times 31 \]

### Step 5: Identify Possible Two-Digit Factors
Now, we need to find pairs of factors of 1612 that are two-digit numbers:
- The factors of 1612 are: 1, 2, 4, 13, 28, 31, 62, 806, 1612.
- The two-digit factors are: 13, 28, 31, 62.

### Step 6: Check for Reverse Order
Now we check which of these pairs have digits that are reverses of each other:
- \( 13 \) and \( 31 \) (reverse)
- \( 28 \) and \( 82 \) (not a factor)
- \( 62 \) and \( 26 \) (reverse)

### Step 7: Verify the Products
Now we verify the products:
1. \( 13 \times 31 = 403 \) (not equal to 1612)
2. \( 62 \times 26 = 1612 \) (this works)

### Step 8: Identify the Greater Number
Among the valid pairs \( (62, 26) \), the greater number is:
\[ \text{Greater Number} = 62 \]

### Final Answer
The greater number is \( \boxed{62} \).

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