x86-64 machine code, 12 bytes

This is a function, callable with args n=RCX, k=ESI. 32-bit return value in EAX.

Callable from C with the x86-64 System V calling convention with dummy args to get the real args into the right registers. uint32_t factk(int, uint32_t k, int, uint64_t n); I couldn't just use Windows x64 because 1-operand mul clobbers RDX, and we don't want REX prefixes to access R8/R9. n must not have any garbage in the high 32 bits so JRCXZ works, but other than that it's all 32-bit.

NASM listing (relative address, machine code, source)

 1                         factk:
 2 00000000 6A01             push 1
 3 00000002 58               pop rax             ; retval = 1
 4 00000003 E306             jrcxz  .n_zero      ; if (n==0) return
 5                         .loop:                ; do {
 6 00000005 F7E1              mul   ecx            ; retval *= n  (clobbering RDX)
 7 00000007 29F1              sub   ecx, esi       ; n -= k
 8 00000009 77FA              ja   .loop         ; }while(sub didn't wrap or give zero)
 9                         .n_zero:
10 0000000B C3               ret

0xc = 12 bytes

Test caller that passes argc as k, with n hard-coded.

align 16
global _start
_start:
  mov  esi, [rsp]
;main:
  mov  ecx, 9
  call factk

  mov  esi, eax
  mov  edx, eax
  lea  rdi, [rel print_format]
  xor  eax, eax
extern printf
  call printf
extern exit
  call exit

section .rodata
print_format: db `%#x\t%u\n`

```