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  • Apr 17, 2008, 07:03 PM
    Siustrulka
    integral factors
    what is the total number of positive integral factors of (60)^5


    A point (x,y) is called integral if both x and y are integers. How many points on the graph of 1/x + 1/y=1/4 are integral points?

    The complex numbers 1+i and 1+2i are both roots of the equation
    x^5- 6x^4+Ax^3+Bx^2+Cx+D=0
    where A,B,C, and D are integers. What is the value of A+B+C+D?
  • Apr 18, 2008, 05:17 AM
    galactus
    1 Attachment(s)
    Quote:

    Originally Posted by Siustrulka
    what is the total number of positive integral factors of (60)^5




    If you mean how many divisors does 60^5 have, then you can make an observation from basic number theory. When you factor a number you get the form . The number of divisors is given by looking at the exponents of the prime factors.
    The number of divisors is (a+1)(b+1)(c+1)...
    So, 60^5 has (11)(6)(6) divisors. A divisor is a number which divides evenly into a number N.


    Quote:

    A point (x,y) is called integral if both x and y are integers. How many points on the graph of 1/x + 1/y=1/4 are integral points?

    How many ways can you find two fractions which sum to 1/4?

    You could solve the equation for y and then plug in integer values of x and see how many return integer values of y.

    Here are a few: (2,-4), (3,-12), (5,20),.

    Quote:

    The complex numbers 1+i and 1+2i are both roots of the equation
    x^5- 6x^4+Ax^3+Bx^2+Cx+D=0
    where A,B,C, and D are integers. What is the value of A+B+C+D?
    This quintic equation has 5 roots. Since 1+i and 1+2i are roots, their conjugates are also roots. So, 4 roots are 1+i, 1+2i, 1-i, 1-2i

    We have one other root we must find since there must be 5 rrots and we have 4.

    Using our given roots we have a quartic:

    This quartic multiplied by the other root, x-y, will give us our given quintic:





    Equate coefficients:











    Now, solve the easy system by solving the first one for y and subbing in the others, then add up your solutions.

    .
  • Apr 23, 2008, 11:32 PM
    bdsatish
    Comment on galactus's post
    Good!

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