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The set of all integral multiples of 5 is a subgroup of
a) The set of all rational numbers under multiplication
b) The set of all integers under multiplication
c) The set of all non-zero rational numbers under multiplication
d) The set of all integers under addition

Answer
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Hint: A subgroup must satisfy all the properties of a group: Closure, Associativity, Identity, and Inverse. A sufficient condition is that it must satisfy the closure property. Note that “The set of all integral multiples of 5” is obtained when we multiply 5 with any number ‘n’.

Complete step-by-step solution:
Example: When we multiply 5 with n=1, we get 5. When we multiply 5 with n=2, we get 10 and so on.
A group is a set G together with an operation (.), which satisfies the following properties: If for aG, bG and cG, we have that
Closure: For aG, bG , we have . bG. This says that for any two elements in G if we apply the operation (.) then the resulting element will also be in G
Associativity: For aG, bG and cG , we have .(b . c) = (a . b).c . This says that for any three elements in G, associativity between them holds when the operation (.) is applied.
Identity: For aG, there exists an element eG such that . e = e . a = a , where e is the identity element.
Inverse: For aG, there exists an element a1  G such that a1= a1 . a = e . Here a1 is the inverse element.
A subgroup must satisfy all the properties of a group. The sufficient condition for a subgroup is that it must satisfy the closure condition.
“The set of all integral multiples of 5” is obtained when we multiply 5 with any number ‘n’. Let H={5n, where nI} . Here ‘I’ is a set with all ‘n’ values. Here H is a non-empty set with at least one element.
Let a=5x and b=5y . Now we apply an addition operator for a and b. So, a+b=5x+5y=5(x+y) . Since 5xG and 5yG , 5(x+y)G . Hence, closure is satisfied.
Hence, the correct answer is
The set of all integral multiples of 5 is a subgroup of
d)The set of all integers under addition.

Note: We can also find the correct answer by applying all the properties of a group to the set of all integral multiples of 5. Example: let a=5, b=10 and c=15 . Now a+b = 5+10=15=5(3)=5(1+2)G , a+(b+c)=5+(10+15)=30 and (a+b)+c=(5+10)+15=30 ,a+e=5+0=5=0+5=e+a , where e=0 , a+a1=5+(5)=0=e and a1+a=(5)+5=0=e , where a1=5
Hence, the set of all integral multiples of 5 satisfies all the properties of a group. This verifies our answer that the set of all integral multiples of 5 is a subgroup of the set of all integers under addition.


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