So the smallest set of integers containing 8 and 14? That would, of course, include 0 as well as 8+ 8= 16 and, in fact, any multiple of 8 or 14. It would include 8+ 14= 22, 14- 8= 6, etc. In fact it would include any x= 8n+ 14m= 2(4n+ 7m) for any integers n and m. (So, first, it must include only even integers.) What can you say about numbers of the form 4n+ 7m? One of the results of the "Euclidean Algorithm" is that since 4 and 7 are "relatively prime", there exist m and n such that 4n+ 7m= x forallintegers x.

8 and 13 are relatively prime.b. <8, 13>

6= 3(2), 15= 3(5) so 6m+ 15n= 3(2m+ 5n).c. <6, 15>

Is no information at all given about m and n?d. <m, n>

[quote]e. <12, 18, 45>.[quote]

12= 2(2)(3), 18= 2(3)(3), 45= 3(3)(5). The greatest common divisor is 3.

You mean "the subgroup is <k>."In each part, find an integer k such that the subgroup is k.