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01 Dec 2025, 22:21
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D
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Difficulty:
25%
(medium)
Question Stats:
77% (01:41) correct
23%
(01:47)
wrong
based on 161
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If \(a\), \(b\), \(c\), and \(d\) are four integers greater than 1, which of the following cannot be a multiple of any of \(a\), \(b\), \(c\), or \(d\)?
A. \(\frac{a }{ b}\) + \(\frac{c }{ d}\)
B. \(\frac{a }{ b}\) – \(\frac{c }{ d}\)
C. \(abcd\) + 2
D. \(abcd\) – 1
E. \(abcd\) + 10
A. \(\frac{a }{ b}\) + \(\frac{c }{ d}\)
B. \(\frac{a }{ b}\) – \(\frac{c }{ d}\)
C. \(abcd\) + 2
D. \(abcd\) – 1
E. \(abcd\) + 10
01 Dec 2025, 22:44
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ExpertsGlobal5
Let us begin by considering Options C and E.
If any one of the integers \(a\), \(b\), \(c\), or \(d\) equals 2, then both \(abcd\) + 2 and \(abcd\) + 10 become even, and therefore are multiples of 2. This means that, under an appropriate choice of values, Options C and E can be multiples of at least one of the four integers. Thus, we eliminate C and E.
Let's take Option D: \(abcd\) – 1.
Since \(abcd\) is the product of the four integers, it is guaranteed to be a multiple of each of \(a\), \(b\), \(c\), and \(d\). For the entire expression \(abcd\) – 1 to be a multiple of any of these integers, the term 1 would also need to be a multiple of at least one among \(a\), \(b\), \(c\), or \(d\). However, the problem explicitly states that all four integers are greater than 1, which makes this impossible because 1 cannot be a multiple of any integer greater than 1.
Option D
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02 Dec 2025, 00:28
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A good way to approach this question is through substitution.
However, remember that multiple options may appear correct depending on the values chosen for a, b, c, and d.
However, remember that multiple options may appear correct depending on the values chosen for a, b, c, and d.
- First, let a, b, c, d = 2, 3, 4, 5.
Substituting these eliminates options C and E. - Now the remaining possibilities are A, B, and D.
To eliminate A and B, substitute equal values such as a = b = c = d = 2.
With this substitution, A and B are eliminated.
