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Andrei [34K]
3 years ago
12

Prove that if a and b are positive integers whose sum is a prime p, their greatest common divisor is 1.

Mathematics
1 answer:
Tasya [4]3 years ago
7 0

Proof by contadiction.

Let assume there exist such positive integers a and b whose sum is a prime number p, that their greatest common divisor is greater than 1.

a,b\in\mathbb{Z^+}\\d=\text{gcd}(a,b)>1\\\\a=de\\b=df\\e,f\in\mathbb{Z^+}\\\\a+b=p\\de+df=p\\d(e+f)=p\\

Since p is a prime number and d>1, then d=p \wedge e+f=1, but we assumed earlier that e,f\in\mathbb{Z^+}, and there are no two positive integers that sum up to 1.

q.e.d.

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5 0
3 years ago
If a graph of y=-4x +2 were changed to a graph of y=-4x+5, how would the y- intercept change?
Dmitry [639]

Answer:

So if y=-4x+2 was changed to y=-4x+5, then the y-intercept would increase by 3.

The y-intercept was (0,2) then it becomes (0,5) in the new line.

Step-by-step explanation:

The slope-intercept form a line is y=mx+b where m is the slope and the y-intercept is b.

Both of these equations given are in this form.

y=-4x+2 when compared to y=mx+b you see that m=-4 and b=2.

Since b=2 then the y-intercept is 2.

y=-4x+5 when compared to y=mx+b you see that m=-4 and b=5.

Since b=5 then the y-intercept is 5.

So if y=-4x+2 was changed to y=-4x+5, then the y-intercept would increase by 3.

8 0
3 years ago
line j is perpendicular to lines m and n. if the products of the slope of all three lines is -8, what is the slope of line j?
jeka94

Answer:

the slope of line j is -1/8, the slope of lines m and n are both 8.

Explanation step by step:

Suppose j is the slope of line j, m the slope of line m and n the slope of line n. Since:

1. line j is perpendicular of line m, so j*m= -1

2. J is perpendicular of line n, so  j*n=-1

3. m and n and parallel so m=n

As we want to know the slope of j we clear the equations presented to us in order to find it.

(1) m* n* j = -8, since n=m we have  n^{2}* j = -8.

we clear j; n^{2} * j = -8 \\ j =\frac{-8}{n^{2} }

replacing in the equation n*j = -1 we get

n*(-8/n^2) = -1. Thus n = 8. Since j = -1/n = -1/8.

7 0
3 years ago
Answer choices: (5,6) (5,2) (3,6) (3,2).
Degger [83]

Answer:

(3,6)

Step-by-step explanation:

If the starting point is the dot then we start from there - (4,4) then we go up 2 units at (4,6) then moving one unit to the left our answer is (3,6).

6 0
3 years ago
Is this triangle congruent? pls answer i will mark you brainlyist :)
tiny-mole [99]

Answer:

no

Step-by-step explanation:

6 0
3 years ago
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