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LCM of Rational Numbers

The least common multiple (LCM) of a set of numbers A is the smallest integer b such that b/a is an integer for all integers a in A. This definition can be extended to rational numbers!

Task

Find the smallest positive rational b such that b/a is an integer for all rationals a in the input.

Rules

Standard loopholes are forbidden.
You may take numerators and denominators separately in the input, but may not take doubles, floats, etc.
The input may not be fully reduced.
You may take integer inputs as rationals with denominator of 1.
Submissions that would feed rational numbers to an LCM/GCD builtin are allowed, but non-competing.

Test Cases

In: 3
Out: 3
In: 1/17
Out: 1/17
In: 1/2, 3/4
Out: 3/2
In: 1/3, 2/8
Out: 1
In: 1/4, 3
Out: 3
In: 2/5, 3
Out: 6
In: 1/2, 3/4, 5/6, 7/8
Out: 105/2

This is code-golf, so submissions using the fewest bytes win!

Answer*

# Common Lisp, 154 bytes

 (defun f(l &aux(s(pairlis l l)))(loop(and(eval`(=,@(mapcar'car s)))(return(caar s)))(let((x(assoc(reduce'min s :key'car)s)))(rplaca x(+(car x)(cdr x))))))

[Algorithm used](https://en.wikipedia.org/wiki/Least_common_multiple#A_simple_algorithm) (specified for integers, but works also for rationals).

First make an associative list of the input data with itself, to get track of the initial values of the elements, so the operating sequence is given by the “car”s of the list.

 (defun f(l &aux (s (pairlis l l))) ; make the associative list
 (loop
 (when (eval `(= ,@(mapcar 'car s))) ; when the car are all equal
 (return (caar s))) ; exit with the first one
 (let ((x (assoc (reduce 'min s :key 'car) s))) ; find the (first) least element
 (rplaca x (+ (car x) (cdr x)))))) ; replace its car adding the original value (cdr)

Test cases:

 CL-USER> (f '(3))
 3
 CL-USER> (f '(1/17))
 1/17
 CL-USER> (f '(1/2 3/4))
 3/2
 CL-USER> (f '(1/3 2/8))
 1
 CL-USER> (f '(1/4 3))
 3
 CL-USER> (f '(2/5 3))
 6
 CL-USER> (f '(1/2 3/4 5/6 7/8))
 105/2

Note: The solution is without the use of the builting `lcm` and `gcd`, that accept integers.

If this is an answer to a challenge…

…Be sure to follow the challenge specification. However, please refrain from exploiting obvious loopholes. Answers abusing any of the standard loopholes are considered invalid. If you think a specification is unclear or underspecified, comment on the question instead.
…Try to optimize your score. For instance, answers to code-golf challenges should attempt to be as short as possible. You can always include a readable version of the code in addition to the competitive one. Explanations of your answer make it more interesting to read and are very much encouraged.
…Include a short header which indicates the language(s) of your code and its score, as defined by the challenge.

More generally…

…Please make sure to answer the question and provide sufficient detail.
…Avoid asking for help, clarification or responding to other answers (use comments instead).

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\$\begingroup\$ W00t? Try this at your REPL (/ (lcm 1 3 5 7) (gcd 2 4 6 8)). \$\endgroup\$

Kaz
– Kaz

2017年06月19日 14:24:36 +00:00
Commented Jun 19, 2017 at 14:24
\$\begingroup\$ @Kaz, since, as it said in problem, "Submissions that would feed rational numbers to an LCM/GCD builtin are allowed, but non-competing". \$\endgroup\$

Renzo
– Renzo

2017年06月19日 14:27:10 +00:00
Commented Jun 19, 2017 at 14:27
\$\begingroup\$ In Lisp terms, strictly speaking, we are in fact feeding rationals when we call (lcm 1 3 5 7), since integers are a subtype of rationals, but I think the rule is supposed to exclude use of a lcm or gcd which allows rational inputs. \$\endgroup\$

Kaz
– Kaz

2017年06月19日 14:34:48 +00:00
Commented Jun 19, 2017 at 14:34
\$\begingroup\$ @Kaz, ops... I misinterpreted the rules! Should I remove the post? (maybe it is not good marketing for Common Lisp :) \$\endgroup\$

Renzo
– Renzo

2017年06月19日 14:52:11 +00:00
Commented Jun 19, 2017 at 14:52
\$\begingroup\$ I'd just put in a note that this is a solution without using the built-in integer lcm and gcd. \$\endgroup\$

Kaz
– Kaz

2017年06月19日 14:55:25 +00:00
Commented Jun 19, 2017 at 14:55

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