Understanding the Problem: Finding the Smaller Number

The problem asks us to find a smaller number, given its relationship with a larger number. We are provided with two key pieces of information:

1. The difference between the two numbers.
2. The result of dividing the larger number by the smaller number (including the quotient and remainder).

Let's define the terms clearly:

• Let the larger number be \( L \).
• Let the smaller number be \( S \).

Based on the problem statement, we can form mathematical equations using these variables.

Formulating Equations from the Given Information

The first piece of information is that the difference between the two numbers is 1564.

Since \( L \) is the larger number and \( S \) is the smaller number, their difference is expressed as:

\( L - S = 1564 \)

This is our first equation.

The second piece of information relates to division. When the larger number \( L \) is divided by the smaller number \( S \), the quotient is 6 and the remainder is 19. The division algorithm states that Dividend = Divisor \(\times\) Quotient + Remainder.

In this case:

• Dividend = \( L \) (the larger number)
• Divisor = \( S \) (the smaller number)
• Quotient = 6
• Remainder = 19

So, we can write the second equation based on the division algorithm:

\( L = 6S + 19 \)

This is our second equation.

Solving the System of Equations to Find the Smaller Number

We now have a system of two linear equations with two variables:

1. \( L - S = 1564 \)
2. \( L = 6S + 19 \)

We want to find the value of the smaller number, \( S \). We can use the method of substitution to solve this system. Equation (2) already gives us an expression for \( L \) in terms of \( S \). We can substitute this expression for \( L \) into Equation (1).

Substitute \( 6S + 19 \) for \( L \) in Equation (1):

\( (6S + 19) - S = 1564 \)

Now, we simplify and solve for \( S \):

Combine the terms with \( S \):

\( 6S - S + 19 = 1564 \)

\( 5S + 19 = 1564 \)

Subtract 19 from both sides of the equation:

\( 5S = 1564 - 19 \)

\( 5S = 1545 \)

Divide both sides by 5 to find \( S \):

\( S = \frac{1545}{5} \)

\( S = 309 \)

The value of the smaller number \( S \) is 309.

Verification of the Solution

Let's check if this value of \( S \) satisfies the original conditions. If \( S = 309 \), we can find \( L \) using the second equation:

\( L = 6S + 19 \)

\( L = 6(309) + 19 \)

\( L = 1854 + 19 \)

\( L = 1873 \)

So, the larger number is 1873 and the smaller number is 309.

Now, let's check the first condition: the difference between the two numbers is 1564.

\( L - S = 1873 - 309 \)

\( 1873 - 309 = 1564 \)

The difference matches the given information.

Next, let's check the second condition: dividing the larger number by the smaller number gives a quotient of 6 and a remainder of 19.

Divide 1873 by 309:

\( 1873 \div 309 \)

We know \( 309 \times 6 = 1854 \).

Subtract 1854 from 1873:

\( 1873 - 1854 = 19 \)

The quotient is 6 and the remainder is 19. This also matches the given information.

Since both conditions are satisfied, our calculated value for the smaller number \( S = 309 \) is correct.

Conclusion: The Smaller Number

By setting up and solving the algebraic equations derived from the problem statement, we found the value of the smaller number.

The smaller number is 309.

Revision Table: Key Steps to Solve Number Problems

Additional Information: Understanding Quotient and Remainder

When an integer is divided by another positive integer, the result can be expressed using a quotient and a remainder. The division algorithm formally states this relationship.

• The Quotient is the whole number of times the divisor goes into the dividend.
• The Remainder is the amount left over after the division is complete.

The remainder must always be less than the divisor and greater than or equal to zero.

Example: When 10 is divided by 3:

• Dividend = 10
• Divisor = 3
• Quotient = 3 (since \( 3 \times 3 = 9 \))
• Remainder = 1 (since \( 10 - 9 = 1 \))

Using the algorithm: \( 10 = 3 \times 3 + 1 \).

In our problem, the divisor is the smaller number \( S \), the quotient is 6, and the remainder is 19. So, \( L = S \times 6 + 19 \), which is \( L = 6S + 19 \). This shows how the division information is converted into an algebraic equation.