EMNA Q.50

50A group of four numbers has only one prime number amongst them. Which of the following must be true about the group?
I. HCF of the four numbers of the group is .either 1 or equal to that prime number.
II. LCM of the four numbers of the group is same as product of the prime number and LCM of the remaining three numbers.
III. Product of four numbers is equal to product of the prime number * HCF of the group * LCM of the group.

Option : A
Explanation : I. In case all the remaining three numbers are multiples of the prime number, HCF of the group is equal to that prime number. otherwise HCF of the group is 1. Thus, I is true.
II. Case 1: Let numbers be 2, 4, 6 and 8, where 2 is only prime number, then LCM = 24.
But LCM of 4, 6 and 8 = 24.
Case 2 : Let numbers be 2, 9, 81 and 27 Then LCM = 162 and LCM of 9, 81 and 27 = 81.
Thus, LCM = 81 * 2 = 162 Hence II is not always true.
III. Let numbers be 2, 4, 6 and 8 LCM =24 and HCF = 2 Product of four numbers = 2 *4 * 6 * 8= 384
Also, 384 ≠ 24 * 2 * 2. Thus, III is not true.

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