Question

Help please...Determine which, if either, of the following sets form a group with respect to the given operation.
I. {1, 2, 3, 4, 5}; multiplication modulo 7
II. the set of integers with respect to multiplication

Answer 1

For the first set, recall that a group must be closed under the given operation. Can you find 2 numbers in the set {1,2,3,4,5} that when multiplied together and reduced modulo 7 are no longer in the set?

Answer 2

no

Answer 3

sorry my computer shut down to install updates. I have never done this kind of math before.

Answer 4

Do you know what reduction modulo 7 is?

Answer 5

nope

Answer 6

is it prime numbers

Answer 7

wow I am lost

Answer 8

It means that you divide by 7, and take the remainder. So 10 modulo 7 is 3, 20 modulo 7 is 6, and 28 modulo 7 is 0. Does this make sense?

Answer 9

no

Answer 10

i divide 10/7= 1.42

Answer 11

only II forms a group
B) only I forms a group
C) neither forms a group
D) both form groups

Answer 12

those are my choices i can pick from

Answer 13

What's the remainder of \(\frac{10}{7}\)? This is just doing the division you learned in elementary school. Don't give me a decimal number.

Answer 14

3

Answer 15

Right. That's why 10 modulo 7 is 3.

Doing a similar thing, 20 modulo 7 is 6. Still with me?

Answer 16

1. 3/7

Answer 17

yeah

Answer 18

Notice that 6 is not in your set {1,2,3,4,5}. Can you find two numbers from that set that multiply to 6?

Answer 19

3,2

Answer 20

Great. But 6 modulo 7, is still 6. Thus, your set is not closed under multiplication modulo 7. Make sense?

Answer 21

a little

Answer 22

so neither forms a group

Answer 23

Correct. However, the second one isn't a group for a different reason. Namely, that there are no multiplicative inverses for the integers.

Answer 24

thank you. I have a few more to do but going to step out for dinner will you be on around 7 tonight?

Answer 25

What time zone?

Answer 26

eastern

Answer 27

I'm not usually on then, but I'll try to log on for at least a little bit.

Answer 28

ok thank you