For how many integers k does the equation [m]x^2-kx+

Question

For how many integers  k  does the equation

x^2-kx+5400=0

have two positive integer roots whose sum is odd but not divisible by 5?

(A) 3
(B) 4
(C) 6
(D) 8
(E) 12

ankushsambare · Answer

Let the positive integer roots be a and b.
Using Vieta’s formulas:
a + b = k
ab = 5400
We need a + b to be odd and not divisible by 5.
Factorize:
5400 = 2^3 × 3^3 × 5^2
For the sum to be odd, one root must be even and the other odd.
So the odd root must be an odd divisor of:
3^3 × 5^2 = 675
Number of odd divisors of 675:
(3 + 1)(2 + 1) = 12
So there are 12 factor pairs with odd sum.
Now remove pairs whose sum is divisible by 5.
Since ab is divisible by 25, at least one factor is divisible by 5.
For the sum to NOT be divisible by 5, exactly one factor must be divisible by 5.
That happens only when one factor contains both 5’s and the other contains none.
So the odd divisor must be:
25 × 3^a, where a = 0, 1, 2, 3
This gives 4 possibilities.
Therefore, the number of possible values of k is 4.
 Answer: B

kevincan · Answer

Here’s a video explanation

Ulfat · Answer

If p and q be the two roots of the given quadratic equation, we have
Sum of roots = k => (p+q) = k.
And, Product of roots = 5400 => (p*q) = 5400,

Now, 5400 =  2^3  x  3^3  x  5^2 , which means we can have maximum (3+1)(3+1)(2+1) = 48 factors.

But p and q must satisfy the following two conditions:

CONDITION 1:  Since, 5400 is an even number it means either p is even or q is even or both p and q are even. 
But, we want p+q to be odd, so both p and q cannot be even. That means one of the factors is even and the other one is odd. Let us consider we want p to be odd and q to be even.

CONDITION 2:  Further, we want that p+q should not be divisible by 5. If some 5s are with p and some 5s are with q, p+q will become divisible by 5. The sum of p+q will not divisible by 5 only when all the 5s are either with p or all the 5s are with q.

Suppose 5400 = p x q, where all the possible values of p are odd so that p+q is odd and p+q is not divisible by 5.
Combining condition 1 and 2, we can have the following values:

p  
  q  
 
  1 
  8 x27x 25  
 
  3 
  8 x9x 25  
 
  9 
  8 x3x 25  
 
  27 
  8 x1x 25  
 
  1x 25  
  8 X27 
 
  3x 25  
  8 x9 
 
  9x 25  
  8 x3 
 
  27x 25  
  8 x1

Thus k (= p+q) has 8 values such that each value is odd and not divisible by 5.

p	q
1	8x27x25
3	8x9x25
9	8x3x25
27	8x1x25
1x25	8X27
3x25	8x9
9x25	8x3
27x25	8x1

For how many integers k does the equation [m]x^2-kx+

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For how many integers k does the equation [m]x^2-kx+

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