Show that the equation 155x -465y 35 has no integer solutions

I mean

Show that the equation 155x -465y=35has no integer solutions

To show that the equation 155x - 465y = 35 has no integer solutions, we can use a method called the "contradiction method" or "proof by contradiction."

First, let's assume that there exist integer solutions for this equation. Let's suppose that both x and y are integers that satisfy the given equation.

We can rearrange the equation by subtracting 35 from both sides:

155x - 465y - 35 = 0

Dividing the entire equation by 5, we get:

31x - 93y - 7 = 0

Now, let's consider the equation modulo 31 (or take the remainder when divided by 31) to see if there are any integer solutions:

(31x - 93y - 7) mod 31 = 0

Simplifying this, we get:

-93y - 7 ≡ 0 (mod 31)

Multiplying both sides by -1, we have:

93y + 7 ≡ 0 (mod 31)

We can now explore the possible remainders of 93y when divided by 31:

For y = 0, 93y + 7 ≡ 7 (mod 31)
For y = 1, 93y + 7 ≡ 100 (mod 31)
For y = 2, 93y + 7 ≡ 193 (mod 31)
For y = 3, 93y + 7 ≡ 286 (mod 31)
and so on...

If we continue this process, we will observe that the remainders of 93y + 7 when divided by 31 repeat in a cycle of length 31.

However, since the remainders can only be integers between 0 and 30 (inclusive), it is impossible for any of these remainders to be equal to 0.

Therefore, there are no integer solutions for the given equation 155x - 465y = 35.

Hence, it has been proven that the equation 155x - 465y = 35 has no integer solutions.

To show that the equation 155x - 465y = 35 has no integer solutions, we can use the method of linear Diophantine equations.

A linear Diophantine equation is an equation of the form ax + by = c, where a, b, and c are integers, and we want to find integer solutions for x and y.

Here's how we can approach this problem:

1. First, observe that both 155x and 465y are divisible by 5 since both numbers end in 5. Therefore, the left-hand side of the equation is divisible by 5.

2. Now, let's look at the right-hand side of the equation, 35. 35 is also divisible by 5.

3. According to a property of linear Diophantine equations, if a linear combination of two integers is divisible by a number, then both of the integers must be divisible by that number.

4. Since 155x - 465y = 35 is divisible by 5, both 155x and 465y must also be divisible by 5.

5. However, it is not possible for both 155x and 465y to be divisible by 5 simultaneously when the constant term, 35, is also divisible by 5. This is because the sum of two numbers divisible by 5 cannot result in a number that is also divisible by 5.

6. Therefore, we conclude that there are no integer solutions for the equation 155x - 465y = 35.

By using the concept of linear Diophantine equations and divisibility, we have shown that the equation 155x - 465y = 35 has no integer solutions.

465 y = 155 x - 35

y = (155/465) x - 35/465

y = (1/3) x - 7/93

when x = 0
y = -7/93

no matter what you do starting with -7/93 going up with a slope of 1/3 it will be hard to get an integer y