GATE2006-3

Question

The set $\{1,2,3,5,7,8,9\}$ under multiplication modulo $10$ is not a group. Given below are four possible reasons. Which one of them is false?

• A. It is not closed
• B. $2$ does not have an inverse
• C. $3$ does not have an inverse
• D. $8$ does not have an inverse

Comments

8*7=56%10=6..not in the set ...so it is not closed ...why it cant be the anssssss

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@asu Bhai question is bit tricky asked

they are asking which one is false.

Here 3 having inverse which is 7 .bcz 3*7=21mod10=1 //1 is identity element

but they are telling 3 does not have an inverse. So false.So C is Ans.

(8 does not have an inverse.This statement is true..but they asked for false. so it cant be ans)

Answer 1

Answer: C

$3$ has an inverse, which is $7.$
$3*7 \mod 10 = 1.$

Comments

The given relation is not closed also, 2*8 mod 10 = 6 which doesn't belong to the set. Can it be a reason for it not being a group?

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yes , it is one of the reasons.

Answer 2

Ans C.

Comments

How??

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In the question it is asked which is false. and given that it's not a group. As we can see 3 have an inverse which is 7 therefore this statement is false

Answer 3

Hi , please correct me if I am wrong. The set is {1,2,3,5,7,8,9} and we need to do multiplication modulo 10.  So, 2 (multiplication modulo 10 ) 2 = 4 , which is not in the set. That means , it is not closed.

Please correct me , if I am wrong .

Comments

You're right 0,4,6 are missing ; so for every n mod 10 = {0,4,6} this set is definitely not closed.

But trick is it is not closed is not false reason ; it's True.

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Ohh yeah.. sorry didn't see the question properly. my bad :( . Yes , it is true. thanks

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Question is asking which of the statement is not FALSE.. Yes it is not closed is a True statement.