Solving for Two Positive Numbers

The problem asks us to find the greater of two positive numbers given two conditions about them. Let's break down the information given.

We are told about two positive numbers. Let's call the greater number G and the smaller number S.

The first condition is that the two positive numbers differ by 1280. This means the difference between the greater number and the smaller number is 1280.

Mathematically, this can be written as:

\(G - S = 1280\)

We can rearrange this equation to express the greater number G in terms of the smaller number S:

\(G = S + 1280\)

The second condition involves dividing the greater number by the smaller number. We are told that when the greater number is divided by the smaller number, the quotient is 7 and the remainder is 50.

Recall the Division Algorithm, which states that for any integers a (dividend) and b (divisor) with b > 0, there exist unique integers q (quotient) and r (remainder) such that \(a = bq + r\), where \(0 \le r < b\).

In our case, the dividend is the greater number G, the divisor is the smaller number S, the quotient \(q\) is 7, and the remainder \(r\) is 50.

So, according to the Division Algorithm, we can write the second condition as:

\(G = 7S + 50\)

Finding the Values of the Numbers

Now we have a system of two linear equations with two variables, G and S:

1. \(G = S + 1280\)
2. \(G = 7S + 50\)

Since both equations are equal to G, we can set the right-hand sides equal to each other to solve for S:

\(S + 1280 = 7S + 50\)

Now, let's solve this equation for S:

• Subtract S from both sides:

\(1280 = 7S - S + 50\)

\(1280 = 6S + 50\)

• Subtract 50 from both sides:

\(1280 - 50 = 6S\)

\(1230 = 6S\)

• Divide by 6 to find S:

\(S = \frac{1230}{6}\)

\(S = 205\)

So, the smaller number is 205.

Now that we have the value of the smaller number S, we can find the greater number G using either of the original equations. Let's use the first equation:

\(G = S + 1280\)

Substitute \(S = 205\) into this equation:

\(G = 205 + 1280\)

\(G = 1485\)

The greater number is 1485.

Verification

Let's check if these two numbers, G = 1485 and S = 205, satisfy both original conditions:

1. Do they differ by 1280?

\(1485 - 205 = 1280\)

Yes, this condition is met.

2. When the greater number (1485) is divided by the smaller number (205), is the quotient 7 and remainder 50?

We need to check if \(1485 = 7 \times 205 + 50\).

Calculate \(7 \times 205\):

\(7 \times 205 = 1435\)

Now add the remainder:

\(1435 + 50 = 1485\)

Yes, this condition is also met, and the remainder 50 is less than the divisor 205 (\(50 < 205\)), as required by the Division Algorithm.

The Greater Number

Based on our calculations and verification, the greater number is 1485.

Revision Table: Key Steps

Additional Information: Systems of Equations and Division Algorithm

This problem is a classic example of how word problems can be translated into a system of algebraic equations. A system of equations involves two or more equations with the same variables. The solution to the system is the set of values for the variables that satisfy all equations simultaneously.

In this case, we had two equations involving G and S. We used the substitution method, where we expressed G in terms of S from one equation and substituted it into the other equation. This reduced the system to a single equation with one variable (S), which we could then solve.

The Division Algorithm is a fundamental concept in number theory. It formally defines the result of division with remainder. For any integer dividend \(a\) and a positive integer divisor \(b\), there's a unique quotient \(q\) and remainder \(r\) such that \(a = bq + r\) and \(0 \le r < b\). This theorem is essential for understanding division and related concepts like modular arithmetic.

In our problem, \(a=G\), \(b=S\), \(q=7\), and \(r=50\). The condition \(0 \le r < b\) means \(0 \le 50 < S\). Since we found \(S=205\), the condition \(50 < 205\) is true, validating our use of the Division Algorithm here.