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TIFR-2014-Maths-A-14

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Let $G$ be a group and let $H$ and $K$ be two subgroups of $G$. If both $H$ and $K$ have $12$ elements, which of the following numbers cannot be the cardinality of the set $HK = \left\{hk : h \in H, k \in K\right\}$? 

  1. $72$
  2. $60$ 
  3. $48$ 
  4. $36$
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I think 60 is not  be the cardinality of the set HK

explanation cardinality of set HK= 144 (H=12 ,K=12)

and subgroup is the divisor of  the group so that 

1:144/72=2

2:144/60=2.4

3:144/36=4

4:144/48=3

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