GCD / LCM Polyglots!

C / C++, 79 78 73 bytes

Thanks to @ETHproductions for saving a byte!

int f(int a,int b){int c,d=a*b;for(;a;b=c)c=a,a=b%a;auto e=.5;c=e?d/b:b;}

C calculates the GCD: Try it online!

C++ calculates the LCM: Try it online!

In C, auto e=.5 declares an integer variable with the auto storage class (which is the default), which is then initialized to 0, whereas in C++11 it declares a double, which is initialized to 0.5. So the variable's value will be truthy in C++ and falsy in C.

The function calculates the GCD with Euclid's algorithm, and the LCM by dividing the product of a and b by the GCD.

Omitting the return statement works at least on GCC. The 78 byte solution below should work with any compiler:

int f(int a,int b){int c,d=a*b;for(;a;b=c)c=a,a=b%a;auto e=.5;return e?d/b:b;}

Jelly / Actually, 2 bytes

00000000: 1e 67                                            .g

This is a hexdump (xxd) of the submitted program. It cannot be tested online because TIO doesn't support the CP437 encoding. @Mego was kind enough to verify that this works on Cygwin, which implements CP437 as intended for Actually.

Jelly: GCD

Jelly uses the Jelly code page, so it sees the following characters.

œg

Try it online!

How it works

œ is an incomplete token and thus ignored. g is the GCD built-in.

Actually: LCM

Actually uses CP 437, so it sees the following characters.

▲g

Try it online!

How it works

▲ is the LCM input. Since g (GCD) requires two integer inputs, it isn't executed.

Actually / Jelly, 3 bytes

00000000: 11 1c 67                                         ..g

This is a hexdump (xxd) of the submitted program.

Try it online!¹

Actually: GCD

Actually uses CP 437, so it sees the following characters.

◄∟g

Try it online!

How it works

     (implicit) Read a and b from STDIN and push them on the stack.
◄    Unassigned. Does nothing.
 ∟   Unassigned. Does nothing.
  g  Pop a and b and push gcd(a,b).
     (implicit) Write the result to STDOUT.

Jelly: LCM

Jelly uses the Jelly code page, so it sees the following characters.

×÷g    

Try it online!

How it works

×÷g  Main link. Left argument: a. Right argument: b

×      Multiply; yield ab.
  g    GCD; yield gcd(a,b).
 ÷     Division; yield ab/gcd(a,b) = lcm(a,b).

Note: The formula gcd(a,b)lcm(a,b) = ab holds because a and b are positive.

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¹ _{TIO actually uses UTF-8 for Actually. Since both the ASCII characters and the CP437 characters 0x11 and 0x1c are unassigned, the program works nevertheless.}