Calculate the Number, Divisors Edition

Question

Inspired by this question over on Math.

Let the prime factorization of a number, n, be represented as: \$P(n) = 2^a\times3^b\times5^c\times\cdots\$. Then the number of divisors of n can be represented as \$D(n) = (a+1)\times(b+1)\times(c+1)\times\cdots\$. Thus, we can easily say that the number of divisors of \2ドルn\$ is \$D(2n) = (a+2)\times(b+1)\times(c+1)\times\cdots\$,
the number of divisors of \3ドルn\$ is \$D(3n) = (a+1)\times(b+2)\times(c+1)\times\cdots\$,
and so on.

Challenge

Write a program or function that uses these properties to calculate \$n\$, given certain divisor inputs.

Input

A set of integers, let's call them \$w, x, y, z\$, with all of the following definitions:

• all inputs are greater than 1 -- \$w, x, y, z > 1\$
• \$x\$ and \$z\$ are distinct -- \$x\ne z\$
• \$x\$ and \$z\$ are prime -- \$P(x)=x, D(x)=2, P(z)=z \text{ and } D(z)=2\$
• \$w\$ is the number of divisors of \$xn\$ -- \$D(xn)=w\$
• \$y\$ is the number of divisors of \$zn\$ -- \$D(zn)=y\$

For the problem given in the linked question, an input example could be \$(28, 2, 30, 3)\$. This translates to \$D(2n)=28\$ and \$D(3n)=30\$, with \$n=864\$.

Output

A single integer, \$n\$, that satisfies the above definitions and input restrictions. If multiple numbers fit the definitions, output the smallest. If no such integer is possible, output a falsey value.

Examples:

(w, x, y, z) => output
(28, 2, 30, 3) => 864
(4, 2, 4, 5) => 3
(12, 5, 12, 23) => 12
(14, 3, 20, 7) => 0 (or some other falsey value)
(45, 13, 60, 11) => 1872
(45, 29, 60, 53) => 4176

Rules:

• Standard code-golf rules and loophole restrictions apply.
• Standard input/output rules apply.
• Input numbers can be in any order - please specify in your answer which order you're using.
• Input numbers can be in any suitable format: space-separated, an array, separate function or command-line arguments, etc. - your choice.
• Similarly, if output to STDOUT, surrounding whitespace, trailing newline, etc. are all optional.
• Input parsing and output formatting are not the interesting features of this challenge.
• In the interests of sane complexity and integer overflows, the challenge number \$n\$ will have restrictions such that \1ドル < n < 100000\$ -- i.e., you don't need to worry about possible answers outside this range.

Related

• Count the divisors of a number
• Give the smallest number that has N divisors
• Divisor sum from prime-power factorization

• Dennis: So, if the smallest solution is larger than 100,000, I can choose to return either a solution or zero?
• AdmBorkBork: @Dennis If it makes your code shorter, sure. Either would be acceptable.

Answer 1

Jelly, (削除) 17 (削除ここまで) 16 bytes

×ばつ€ȷ5R¤ÆDL€€Z=Ḅi3

This is a brute force solution that tries all possible values up to 100,000. Try it online!

Non-competing version

The latest version of Jelly has a bug fix that allows to golf down the above code to 15 bytes.

×ばつ3ドルÆDL€€=Ḅi3

Try it online!

How it works

×ばつ€ȷ5R¤ÆDL€€Z=Ḅi3 Main link. Left input: x,z. Right input: w,y
 ¤ Combine the two atoms to the left into a niladic chain.
 ȷ5 Yield 100,000 (1e5).
 R Apply range. Yields [1, ..., 100,000].
x€ Multiply each r in the range by x and z.
 This yields [[x, ..., 100,000x], [z, ..., 100,000z]].
 ÆD Compute the divisors of each resulting integer.
 L€€ Apply length to each list of divisors.
 This counts the divisors of each integer in the 2D array.
 Z Zip; group the divisors of kx and kz in pairs.
 = Compare each [divisors(kx), divisors(kz)] with [w, y].
 This yields a pair of Booleans.
 Ḅ Convert each Boolean pair from binary to integer.
 i3 Find the first index of 3. Yields 0 for not found.

• AdmBorkBork: Congrats, you win by default! :D