Problem 2

The equation is actually $3x+y=35$ . Divide it by 3 so that the coefficient of x is 1. Rearrange it and you can get $x-11=\frac{2-y}{3}$ Because $x$ and $y$ are integers, $x-11$ is also an integer, then $\frac{2-y}{3}$ must be also an integer.

Suppose $\frac{2-y}{3}=k$ where $k$ is an integer, then $y=2-3k$ . Because $y$ is positive, so $k<1$ .

Substitue $k$ in $3x+y=35$ you will get $x=11+k$ . Because $x>0$ , so $k>-11$ , hence $-11<k<1, k=-11,-10,-9, ... , 0$ There are only 11 solutions for $k$ , so the answer is 11.

3x+y=35, so, for y to be non-zero, positive, 3x<35, and for x to be non zero, positive, x>0 => o<x<12, thus x can be anything from 1 to 11, which accounts to 11 total values.

Because x and y are both non zero integers, they must have a value of at least one. Therefore, we can subtract 8 from 70 to create the new equation

$6x+2y=62$ , which allows for x and y to be 0.

Then, we can notice that in our new equation, for any value of x, y must be at least 1, so that the remainder will be a multiple of 6. Therefore, we can create the new equation

$6x+2y=60$

Now we need only look at the possible values of 6 (since 2 is a multiple of 6, the space that is leftover will be filled up with 2s: for every possible value of x, there exists a corresponding value for y). At most, x=10, and at its minimum, x is 0. Therefore, all values possible for x are inclusively in between 0 and 10. 10-0+1= 11. There are 11 possible values for x, therefore there are 11 ordered pairs.

Both multiples of 6 and 2 are all even. Since the ordered pairs are non-zero positive integers, we can multiply 6 by 1 up to 11 while the multiples of 2 will just complement the multiples of 6, summing up to 70. Hence, the answer is 11.

6x + 2y = 70 or, 3x + y = 35

y is alone here. So we leave that in peace, and try to match up the pairs of x. It easy to see that we can have 11 different choices, since (11)(3) = 33. This is the maximum we can reach, and then add up a y = 2 to balance up the 35.

Hence we have our answer: 11

The thing to understand here, x plays a major role here than y, y merely does the balancing act.

6x + 2y = 70 ⇒ 3x + y = 35 ⇒ y=35-3x ⇒ 35-3x>0 ⇒ 35 - 3x > 0 .(-1) ⇒ 3x - 35 < 0 ⇒ x<11,6... ∴ 0 < x ≤ 11