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# If a and b are integers such that a > b > 1, which of the following ca

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Math Expert
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If a and b are integers such that a > b > 1, which of the following ca  [#permalink]

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13 Aug 2018, 03:55
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55% (hard)

Question Stats:

53% (01:12) correct 47% (01:07) wrong based on 144 sessions

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If a and b are integers such that a > b > 1, which of the following cannot be a multiple of either a or b?

(A) a – 1
(B) b + 1
(C) b – 1
(D) a + b
(E) ab

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Re: If a and b are integers such that a > b > 1, which of the following ca  [#permalink]

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13 Aug 2018, 06:35
1
Bunuel wrote:
If a and b are integers such that a > b > 1, which of the following cannot be a multiple of either a or b?

(A) a – 1
(B) b + 1
(C) b – 1
(D) a + b
(E) ab

Suppose a=4; b=3.
4>3>1
A) a-1=3--multiple of 3 --eliminate
B) b+1=4--multiple of 4 --eliminate
C) b-1=2--not a multiple -- HOLD
D) a+b=7--not a multiple -- HOLD
E) ab=12--multiple of 3 and 4 --eliminate
Now let us check some other values of a,b
a=6,b=3; a+b=9 --multiple of 3 --eliminate
but b-1=2 --not a multiple - correct answer.
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Re: If a and b are integers such that a > b > 1, which of the following ca  [#permalink]

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15 Aug 2018, 20:59
Bunuel wrote:
If a and b are integers such that a > b > 1, which of the following cannot be a multiple of either a or b?

(A) a – 1
(B) b + 1
(C) b – 1
(D) a + b
(E) ab

Given, a and b are integers & a > b > 1 implies that a and b are positive integers.

Note:- A positive multiple is always greater than or equal to the number it's a multiple of.
So, the answer option which violates our noted reasoning would be the correct answer to the question stem. (keeping in mind that a > b > 1)

Among answer options, b-1 is smaller than both a and b.

Ans. (C)
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Re: If a and b are integers such that a > b > 1, which of the following ca  [#permalink]

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16 Aug 2018, 01:32
1

Solution

Given:
• a and b are integers
• a > b > 1

To find:
• Which of the given answer choices cannot be a multiple of either ’a’ or ‘b’

Approach and Working:
• For a number to be a multiple of ‘a’, it must be greater than ‘a’
• Similarly, for a number to be a multiple of ‘b’, it must be greater than ‘b’
• a > b > 1, and a, b are integers, implies that a and b are positive integers greater than 1.
• a – 1 is less than a, but it can be greater than b.
o Thus, it can be a multiple of b
• b + 1 is always greater than b.
o Thus, it can be a multiple of b
• b – 1 is less than both, a and b.
o Thus, it cannot be a multiple of both a and b
• a + b is greater than both a and b.
o Thus, it can be a multiple of both a and b
• ab is greater than both a and b.
o Thus, it can be a multiple of both a and b

Hence, the correct answer is option C

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Re: If a and b are integers such that a > b > 1, which of the following ca  [#permalink]

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28 Oct 2018, 19:34
Hi All,

We're told that A and B are integers such that A > B > 1. We're asked which of the following CANNOT be a multiple of either A or B. This question can be solved with some basic concept knowledge of the rules behind 'multiples.'

With the exception of "0", multiples are equal to - or greater than - their base number. For example, the multiples of 2 are 0, 2, 4, 6, 8, 10, 12, etc. Here, we're told that A is GREATER than B - and both of those integers are GREATER than 1. Thus, whatever A and B are, a number that is between B and 1 can NEVER be a multiple of either of those 2 variables. Looking at the answer choices, you should be able to immediately spot a value that is LESS than B.

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Re: If a and b are integers such that a > b > 1, which of the following ca  [#permalink]

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29 Oct 2018, 03:16
Bunuel wrote:
If a and b are integers such that a > b > 1, which of the following cannot be a multiple of either a or b?

(A) a – 1
(B) b + 1
(C) b – 1
(D) a + b
(E) ab

$$a > b \ge 2\,\,\,{\rm{ints}}$$

$$?\,\,:\,\,\underline {{\rm{not}}} \,\,{\rm{multiple}}\,\,{\rm{of}}\,\,a,b\,$$

$${\rm{Take}}\,\,\left( {a,b} \right) = \left( {3,2} \right)\,\,\,\,\left\{ \matrix{ \,a - 1\,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,b\,\,\,\, \Rightarrow \,\,\,\,\left( A \right)\,\,\,{\rm{out}} \hfill \cr \,b + 1\,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,a\,\,\,\, \Rightarrow \,\,\,\,\left( B \right)\,\,\,{\rm{out}} \hfill \cr \,ab\,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,a,b\,\,\,\, \Rightarrow \,\,\,\,\left( E \right)\,\,\,{\rm{out}} \hfill \cr} \right.$$

$${\rm{Take}}\,\,\left( {a,b} \right) = \left( {4,2} \right)\,\,\,\,\left\{ {\,a + b\,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,b\,\,\,\, \Rightarrow \,\,\,\,\left( D \right)\,\,\,{\rm{out}}} \right.$$

Conclusion: the correct answer is (C), by exclusion.

Important: from the fact that b-1 is a POSITIVE integer less than both a and b, we are sure b-1 is not a multiple of any one of them!
(-2 is less than both 1 and 2, and -2 is a multiple of both of them. Be careful not to make wrong conclusions!)

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: If a and b are integers such that a > b > 1, which of the following ca  [#permalink]

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30 Oct 2018, 18:16
Bunuel wrote:
If a and b are integers such that a > b > 1, which of the following cannot be a multiple of either a or b?

(A) a – 1
(B) b + 1
(C) b – 1
(D) a + b
(E) ab

Since b - 1 is less than both a and b, it can’t be a multiple of either a or b.

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Re: If a and b are integers such that a > b > 1, which of the following ca &nbs [#permalink] 30 Oct 2018, 18:16
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