Question

Find the smallest positive integer in the given set. [Hint: Theorem 1.2.]

(a) $\{6u+15v|u,v\in \mathrm{\mathbb{Z}}\}$

(b)$\{12r+17s|r,s\in \mathrm{\mathbb{Z}}\}$

Short answer

1. The smallest positive integer of the given set is 3.
2. The smallest positive integer of the given set is 1.

Step-by-step solution

Theorem

Assume that, a and b are any non-zero integers. Let d be the greatest common divisor. Then $d=au+bv$, where u, vare any integers.

Check for part (a)

It is given that, $\{6u+15v|u,v\in \mathrm{\mathbb{Z}}\}$.

Hence, the numbers are 6 and 15, whose greatest common divisor is 3.

Then the smallest positive integer in the set of all linear combinations of a, b is exactly $\left(a,b\right)$.

$\left(6,15\right)=3$

Check for part (b)

It is given that, $\{12r+17s|r,s\in \mathrm{\mathbb{Z}}\}$.

Hence, the numbers are 12 and 17, whose greatest common divisor is 1.

Then the smallest positive integer in the set of all linear combinations of a, b is exactly $\left(a,b\right)$.

$\left(12,17\right)=1$