x is a positive integer that satisfies x2 – kx + 16 = 0

[bhumika.k.shah]

If k is a constant positive integer, and if x is a positive integer that satisfies x2 - kx + 16 = 0, what is the value of x?

I.k is even
II.k > 9

Source : KNEWTON CAT

I wanna know a quicker approach,if any. Also the difficulty level.

[eaakbari]

IMO E

[bhumika.k.shah]

Hi eaakbari,

could you also explain how you got the answer. Coz for me thats more important.

IMO E

[eaakbari]

Sure I just wanted to know the OA to be sure and not give a wrong explanation.

Stem
We know we have two positive roots due to the signs in the quadratic equation. Remember we need one distinct answer. The equation is such that it can have only three sets of roots
(16,1) and (8,2) and (4,4)

Statement one
This implies its the set of roots (8,2) since their sum will be even but we have two roots and no distinct value of x. Hence Insuff

Statement two
This does not help muchas it could be both the sets.

Combined
We are still unable to come to 1 distinct root
Hence infsuff


Answer E

[neoreaves]

IMO C
what we need to know x = ?

we know that x >0 and an integer and k>0 and an integer

x2 - kx + 16 = 0
x2 - kx = -16
x(x-k) = -16

1) k is even


possibilities

x = 16 then x-k = -1 which is only possible if k = 17 not an even integer so not possible
x = 8 then x-k = -2 which is only possible if k = 10 -- Possible
x = 4 then x -k = -4 which is only possible if k = 8 -- Possible
x = 1 then x - k = -16 which is only possible if k = 17 ..not even so not possible

Two possibilities so insufficient

2) k > 9

from the above analysis there are more than 1 values of x that satisfy this condition

c) the only possibility is when k = 10 which gives x = 8 thus Sufficient

[eaakbari]

x^2 - kx + 16 will have to have 2 positive roots

If their product that is ab = 16 a positive quantity and their sum (or difference) being a negative quantity implies the equation is of the form

(x-a)(x-b)

And in our case ab = 16

[gmatmachoman]

@bhumika,

To solve x in a quadriatic equation of the form (ax^2 +b.x+c=0), we need to know the values of a,b,c.

Here coefficient of X term is K(x^2 - kx + 16 ).

To find value of x, we need to know value of K.

Does st 1 gives the value of K??

k is even ----

Infinite number of options are possible--Insufficient

Does st 2 help us in getting value of K?

k > 9

Again K could be anything

So insufficient

Combining St 1& St 2: still K value could be anything

So E.

[neoreaves]

bhumika ....can you please post the OA ....

[bhumika.k.shah]

[spoiler]We are given the quadratic equation x^2 - kx + 16 = 0 and asked to find value of x, given that k and x are positive integers, and that k is a constant.

Statement 1 tells us that k is even. We don't know the value of k, but we can try factoring x^2 - kx + 16 in different ways to see what values of k (and x) are possible. To do this, we find pairs of factors of 16, since the constant term in the quadratic expression is 16: (1,16), (2, 8), (4, 4). This means that we can factor the expression as follows:

(x - 1)(x - 16), which gives us x^2 - 17x + 16. In this case, k is odd (17), and x = 1 or x = 16.
(x - 2)(x - 8), which gives us x^2 - 10x + 16. In this case, k is even (10), and x = 2 or x = 8.
(x - 4)(x - 4), which gives us x^2 - 8x + 16. In this case, k is even (8), and x = 4.
k is even in cases 2 and 3. In the case 2, x can be 2 or 8. In case 3, x must be 4. Since x can have three different values, Statement 1 is insufficient. The answer is choice B, C or E.

Statement 2 tells us that k > 9. We can see from the list above that this means that k is either 10 or 17. Depending on the value of k, x could be 1, 2, 8, or 16. Statement 2 is also insufficient. The answer is choice C or E.

Together, we know that k is even and greater than 9. From the list above, we can see that k = 10. However, this only gives us the value of k.

When k = 10, x = 2 or x = 8. Since we still cannot narrow down x to one possible value, the statements together are not sufficient.[/spoiler]

Answer choice E is correct.

bhumika ....can you please post the OA ....

[gmatmachoman]

We are given the quadratic equation x^2 - kx + 16 = 0 and asked to find value of x, given that k and x are positive integers, and that k is a constant.

Statement 1 tells us that k is even. We don't know the value of k, but we can try factoring x^2 - kx + 16 in different ways to see what values of k (and x) are possible. To do this, we find pairs of factors of 16, since the constant term in the quadratic expression is 16: (1,16), (2, 8), (4, 4). This means that we can factor the expression as follows:

(x - 1)(x - 16), which gives us x^2 - 17x + 16. In this case, k is odd (17), and x = 1 or x = 16.
(x - 2)(x - 8), which gives us x^2 - 10x + 16. In this case, k is even (10), and x = 2 or x = 8.
(x - 4)(x - 4), which gives us x^2 - 8x + 16. In this case, k is even (8), and x = 4.
k is even in cases 2 and 3. In the case 2, x can be 2 or 8. In case 3, x must be 4. Since x can have three different values, Statement 1 is insufficient. The answer is choice B, C or E.

Statement 2 tells us that k > 9. We can see from the list above that this means that k is either 10 or 17. Depending on the value of k, x could be 1, 2, 8, or 16. Statement 2 is also insufficient. The answer is choice C or E.

Together, we know that k is even and greater than 9. From the list above, we can see that k = 10. However, this only gives us the value of k.

When k = 10, x = 2 or x = 8. Since we still cannot narrow down x to one possible value, the statements together are not sufficient.

Answer choice E is correct.

bhumika ....can you please post the OA ....

[kstv]

This is outside the scope of GMAT, but still
the eq ax²+bx+ c = 0 will have real roots if
b²-4ac >= 0
x²-kx+16 = 0
since in GMAT all nos are assumed real nos
so k²- 4*16 >= 0
easier to find the correct answer choice