If M and N are positive integers that do not share any factor greater than 1, which of the following statements must be true?I. The least common multiple of M and N has four factorsII. M and N have opposite even-odd natureIII. M = N + 1a)I onlyb)II onlyc)III onlyd)I, II and IIIe)None out of I, II and IIICorrect answer is option 'E'. Can you explain this answer?

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a)
I only
b)
II only
c)
III only
d)
I, II and III
e)
None out of I, II and III

Correct answer is option 'E'. Can you explain this answer?

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Test: LCM And GCD- 2

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If M and N are positive integers that do not share any factor greater ...

Given:

Positive integers M and N
GCD (M, N) = 1
- As GCD(M, N) = 1, that means M and N do not share any common prime factors

To Find: Which of the 3 statements are true for ALL values of M and N? (A must be true statement is true for ALL values of M and N. If even one value of M and N doesn’t satisfy a statement, then it is not a must be true statement)

Approach:

,We will evaluate each of the given 3 statements to determine which is a must be true statement

Working out:

Evaluating Statement I
The least common multiple of M and N has four factors

Since GCD(M,N) = 1, LCM(M,N) = M*N
If the product M*N is of the form P *P or of the form P , where P₁ and P₂ are prime numbers, then Statement I will hold true
- For example, when M = 2 and N = 5
- Or when M = 1 and N = 2
- (Note: a case like M = 2 and N = 2 is not possible because then GCD(M,N) will not be 1)
But for other values of M and N, Statement I will not hold true.
- For example, when M = 3 and N = 2
So, Statement I is not a must be true statement

Evaluating Statement II
M and N have opposite even-odd nature
?The fact that GCD(M, N) = 1 indicates that they are not both even (because if M and N were both even, then they would both be divisible by 2. So, their GCD would have been 2 or another even number then)
However, it does not necessarily mean that M and N have opposite even-odd nature
- M and N may have even-odd nature. Example, M = 8 and N = 9
- Or, M and N may be both odd. Example, M = 19 and N = 21
- So, Statement II is not a must be true statement

Evaluating Statement III
M = N + 1
- For any pair of consecutive numbers, the GCD is equal to 1
- But the vice-versa is not true. So, if GCD(M,N) = 1, this does not necessarily mean that M and N are consecutive integers. For example, M could be 19 and N could be 21
- Thus, Statement III is not a must be true statement

Looking at the answer choices, we see that the correct answer is Option E

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If M and N are positive integers that do not share any factor greater than 1, which of the following statements must be true?I. The least common multiple of M and N has four factorsII. M and N have opposite even-odd natureIII. M = N + 1a)I onlyb)II onlyc)III onlyd)I, II and IIIe)None out of I, II and IIICorrect answer is option 'E'. Can you explain this answer?

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