[Siustrulka]

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?

[galactus]

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.

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),.

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.

.

[bdsatish]

Good!