# How to find the smallest integer in subgroup of Z

What is the smallest positive integer (i) in $7\mathbb{Z}\cap9\mathbb{Z}$? (ii) $7\mathbb{Z}+9\mathbb{Z}$? where $\mathbb{Z}$ is the set of integers.

I said that $7\mathbb{Z}\cap9\mathbb{Z}=\mathbb{Z}$ but then I can't conclude, the smallest integers do not really exists no? For $7\mathbb{Z}+9\mathbb{Z}$, said that it was equal to $63\mathbb{Z}$, but again there, there is an infinite number of numbers in $63\mathbb{Z}$, how can I find the smallest? It seems like I don't really get the question...

• $7\Bbb Z\cap 9\Bbb Z = \{x\in \Bbb Z~:~7\mid x~\text{and}~9\mid x\}$. Numbers that are simultaneously divisible by $7$ and divisible by $9$ can be characterized as being divisible by some number. What number? Think about greatest common divisors. Meanwhile $7\Bbb Z+9\Bbb Z = \{x+y~:~7\mid x~\text{and}~9\mid y\}$ – JMoravitz Dec 21 '17 at 2:07
• Another comment., "there is an infinite number of numbers in $63\Bbb Z$, how can I find the smallest?" You are correct, there are, and no you can not find the smallest, it is unbounded. However! You can find the smallest positive such number by the well ordered principle. Note that $63\Bbb Z = \{\dots,-126,-63,0,63,126,\dots\}$. It should be clear what the smallest positive number in it is. – JMoravitz Dec 21 '17 at 2:43

I think you're a little confused on the definitions here.

What does it mean for $x$ to be in $7 \mathbb{Z} \cap 9 \mathbb{Z}$? It means that $x$ is both a multiple of 7 and a multiple of 9. What's the smallest positive multiple of both 7 and 9 you can think of?

As for $7 \mathbb{Z} + 9 \mathbb{Z}$: any element in here is of the form $7a + 9b$, for integers $a$ and $b$. There is a way to find the smallest positive integer that can be expressed as a linear combination of two integers, have you encountered this before?

• Thank you so much. You made it so much more clear. I think it is the gcd/hcf of 7 and 9 that I can find using Euclid method? – Student number x Dec 21 '17 at 2:47
• yes, that's exactly right – emma Dec 21 '17 at 2:51

No. The smallest positive integer in $a\mathbf Z\cap b\mathbf Z$ is just the l.c.m. of $a$ and $b$, and when $a$ and $b$ are coprime, it is nothing else but their product.

As to $a\mathbf Z+ b\mathbf Z$, it is generated by $\gcd(a,b)$.