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Special Ordered Pairs (Posted on 2007-12-28)

Call the ordered pairs (x,y), where x, y are positive integers such that x*y=A and y is a multiple of x, "special ordered pairs" of A.

1) Find an expression for the number of special ordered pairs of a given A.

2) Show that:
π_i|Ad(i) = π (d(x_k)*d(y_k))^{2^p} with p substituting for n(y_k / x_k),
if (x_k,y_k) for {k=1,2,..} are all possible special ordered pairs of A.

Note: i|A means i is a divisor of A, d(i) is the number of positive divisors of i, n(i) is the number of prime divisors of i and π determines the product

Submitted by Praneeth

Rating: 4.0000 (1 votes)

Solution: (Hide)

1) Let z be the maximum square divisor of A, then no. of Special-ordered pairs of A = d(√z).
2) π d(i) = π (d(x_k)*d(y_k))^{2^n(y_k/x_k)}; where k varies from 1 to d(√z).
where n(x): No. of prime divisors of x and (x_k,y_k) are Special-Ordered pairs of A.
Explanation:
π d(i) {i: every positive divisor of A} = π d(i)*d(A/i) {i: every positive divisor of A less than √A}
Here x*y=A => LCM(x,y)*GCD(x,y)=x*y=A.
We can say that LCM and GCD are also ordered-pairs of A. Special-ordered pairs are nothing but possible (G,L) ordered pairs of A.
Every ordered pair of A has an (L,G) pair which is also an ordered pair of A. We know for an ordered pair (x,y); d(x)*d(y)=d(L)*d(G)=d(x_k)*d(y_k)
We have to find no. of positive integer solutions for (x,y) whose LCM is L and GCD is G; which is 2 ^n(L/G).
Combining all these results, we get the above result.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
re	Donald	2019-05-21 03:44:24
Heuristics for part 1	Charlie	2007-12-28 18:12:47

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