Detailed SolutionStep-I: Given InfoWe are given two positive integers a and b such that their highest common factor is 5. We are asked to find the values of a and b.

Step-II: Interpreting the Question StatementSince, the highest common factor of a and b is 5, it implies that the GCD of (a, b) = 5. So, we can deduce that both a and b definitely have 5 as one of their prime factors. Let’s use the information given in the statements to see if we can find out the values of a and b.

Step-III: Statement-IStatement-I tells us that the LCM (a, b) is the product of one of the integers of a and b with GCD (a, b) i.e. 5. Since, we know that the product of the numbers is equal to the product of their LCM & GCD, we can write

LCM * GCD = a * b⇨ 5 * (a or b) * 5 = a * b

Simplifying this statement would give us the value of one of the number as 25. Since, we do not have any information about the value of the second number, statement-I is insufficient.

Step-IV: Statement-IIStatement-II tells us that the smaller of the two integers a and b is divisible by 4 factors and has the smallest odd prime number (i.e. 3) as its factor. So, the integer can be represented as:

x * y or x^3.

Since, we know that both the integers have 5 as one of its prime factors (since 5 is the GCD) so, we can represent the integer as product of two prime factors i.e. x * y = 3 * 5= 15.

Now, we know that one of the integers has the value of 15, but we are not given any information about the value of the second integer.

Hence, Statement-II is insufficient to answer the question.

Step-V: Combining Statements I & IIStatement- I tells us that one of the integers of a and b has the value as 25. Statement-II tells us that the other integer has the value as 15. This would mean that:

Either a= 25 and b =15 or

a = 15 and b =25.

Hence, we can’t say with certainty the specific values of a and b.

Thus, combining St-I & II is also not sufficient to answer the question.

Hence, the answer is

Option E.Key Takeaways1. The prime factors of the GCD of a set of numbers would also be the prime factors of the numbers themselves.

2. Familiarize yourself with all the names by which the test makers can call the GCD and the LCM.

For example,

• GCD is also known as the HCF

• GCD can also be described as ‘the largest number which divides all the numbers of a set’

• LCM of a set of numbers can also be described as ‘the lowest number that has all the numbers of that set as it factors’

3. Read the question carefully Harley1980 I hope you realized where did you miss

Regards

Harsh

since both integers have 5 as their prime factors, and statement 2 says that smaller integer has 3 as its prime factor, we can say one integer is 5 * 3 = 15

now, can you please tell me why to assume that the smaller integer doesn't have any other prime factors other than 3 and 5..?