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28 May 2023, 04:27
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C
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x and y are distinct positive integers. What is the positive difference between the total factors of x and the total factors of y?
(A) x and y both have exactly two prime factors, i.e. 3 and 7
(B) Both x and y are less than 100.
(A) x and y both have exactly two prime factors, i.e. 3 and 7
(B) Both x and y are less than 100.
28 May 2023, 07:38
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ChandlerBong
Given: x and y are distinct positive integers
We can start with statement 2 as its the easier of the two statements.
Statement 2
(B) Both x and y are less than 100
As we need the number of factors, we need to know the exact values of x and y. Knowing the range of possible values will not help us find the total number of factors as there can be multiple values within that range.
Ex.
Case 1:\( x = 2 * 5\) ; \(y = 3 * 5\\
\)
The number of factors of both x and y is 4
Case 2: \(x = 2^2 * 5\) ; \(y = 3 * 5\)
The number of factors of x is 6, and the number of factors of y is 4. The positive difference is 2. Hence, the statement alone is not sufficient. We can eliminate B and D.
Statement 1
(A) x and y both have exactly two prime factors, i.e. 3 and 7
This statement is also not sufficient, the reasoning is very similar to that of Statement 2. We know that x and y are composed of 3s and 7s, but without any constraint, we can be multiple such values of x and of y.
Ex.
Case 1:\( x = 3^3 * 5\) ; \(y = 3 * 5\\
\)
The number of factors of x is 8, and the number of factors of y is 4. The positive difference is 4.
Case 2: \(x = 3^2 * 5\) ; \(y = 3 * 5\)
The number of factors of x is 6, and the number of factors of y is 4. The positive difference is 2. Hence, the statement alone is not sufficient. We can eliminate A.
Combined
As both values need to be less than 100, the only possible combination is when one value is \(3^2 * 7\) and the other value is \(3 * 7\). Hence, the statements combined is sufficient.
Option C
05 Aug 2024, 00:19
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