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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # If a, b, and c are three positive integers, each greater than 1, what

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Math Expert V
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If a, b, and c are three positive integers, each greater than 1, what  [#permalink]

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If a, b, and c are three positive integers, each greater than 1, what is the remainder when the product abc is divided by 2?

(1) If each of a, b, and c is divided by 2, the product of all three remainders is 0.

(2) If each of a, b, and c is divided by 2, the sum of all three remainders is 2.

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Re: If a, b, and c are three positive integers, each greater than 1, what  [#permalink]

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1
Quote:
If a, b, and c are three positive integers, each greater than 1, what is the remainder when the product abc is divided by 2?

(1) If each of a, b, and c is divided by 2, the product of all three remainders is 0.

(2) If each of a, b, and c is divided by 2, the sum of all three remainders is 2.

Question: Remainder [a*b*c/d] = ?

CONCEPT: When a number is divided by 2 then
- remainder will be 0 if number (a*b*c) in this case is even
- remainder will be 1 if number (a*b*c) in this case is odd

Statement 1: If each of a, b, and c is divided by 2, the product of all three remainders is 0.

Rule: Remainder [a*b*c/d] = Remainder {Remainder [a/d] * Remainder [b/d] * Remainder [c/d]}/2

Since product of all three remainders when a, b and c are individually divided by 2 is zero, therefore, Remainder when a*b*c is divided by 2 will also be Zero

SUFFICIENT

Statement 2: If each of a, b, and c is divided by 2, the sum of all three remainders is 2

Rule: Remainder [a+b+c/d] = Remainder {Remainder [a/d] + Remainder [b/d] + Remainder [c/d]}/2

Since sum of three remainder = 0 which remainder can only be {0 and 1}
therefore, those three remainder will be {1, 1 and 0}
i.e. One of the remainders when a, b and c are divided by 2 = 0

therefore remainder when product a*b*c when divided by 2 is also ZERO

SUFFICIENT

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Re: If a, b, and c are three positive integers, each greater than 1, what  [#permalink]

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If a, b, and c are three positive integers, each greater than 1, what is the remainder when the product abc is divided by 2?

(1) If each of a, b, and c is divided by 2, the product of all three remainders is 0.
If the product of remainders is 0, then we can infer that there is at least one even integer (either from a, b or c)
Anything multiply by an even integer will result in an even integer.
So, when a*b*c (even) will be divided by 2, the remainder will always be ZERO
Sufficient. A D / B C E

(2) If each of a, b, and c is divided by 2, the sum of all three remainders is 2.
If the sum of the remainders is 2, we can infer that there are 2 Odd Integers and 1 Even integer (Remainder 1+ Remainder 1+ Remainder 0 = Remainder 2)
So, we will get the situation in which we have at least 1 Even integer and anything multiply by an even integer will result in an even integer.
So, when a*b*c (even) will be divided by 2, the remainder will always be ZERO
Sufficient.

'D' is the winner.

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Re: If a, b, and c are three positive integers, each greater than 1, what  [#permalink]

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target find the remainder when integers abc product is divided by 2;

#1

If each of a, b, and c is divided by 2, the product of all three remainders is 0.
this means that either of a,b,c one of the terms is even ; so in that case the product abc will be even ; hence the remainder would be 0 when divided by 2 ; sufficient

#2
If each of a, b, and c is divided by 2, the sum of all three remainders is 2.
this means that either one of the terms of abc is even and other two are odd ; again the product of abc will be even and the remainder would be 0
sufficient
IMO D

If a, b, and c are three positive integers, each greater than 1, what is the remainder when the product abc is divided by 2?

(1) If each of a, b, and c is divided by 2, the product of all three remainders is 0.

(2) If each of a, b, and c is divided by 2, the sum of all three remainders is 2.
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Re: If a, b, and c are three positive integers, each greater than 1, what  [#permalink]

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1
Quote:
If a, b, and c are three positive integers, each greater than 1, what is the remainder when the product abc is divided by 2?

(1) If each of a, b, and c is divided by 2, the product of all three remainders is 0.
(2) If each of a, b, and c is divided by 2, the sum of all three remainders is 2.

abc = even, then remainder even/2 is 0
abc = odd, then a,b,c must all be odd, thus remainder odd/2 is 1

(1) sufic
at least one of the integers is even, thus, abc is even, remainder is 0

(2) sufic
one of the integers is not odd, thus, abc is even, remainder is 0

Ans (D)
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Re: If a, b, and c are three positive integers, each greater than 1, what  [#permalink]

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1
If a, b, and c are three positive integers, each greater than 1, what is the remainder when the product abc is divided by 2?

Product abc would result either in '1' or '0' depending whether all are odd or at least one is even.

(1) If each of a, b, and c is divided by 2, the product of all three remainders is 0.
This implies that at least on is even and even if one is even the product abc would result in remainder as '0'.

SUFFICIENT.

(2) If each of a, b, and c is divided by 2, the sum of all three remainders is 2.
Here again one of the integers is even whose remainder is '0' and other two are odd such that they both have '1' as remainder and sum as '2'.
And even if one is even the product abc is even to result in a remainder in '0'.

SUFFICIENT.

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Re: If a, b, and c are three positive integers, each greater than 1, what  [#permalink]

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1
If a, b, and c are three positive integers, each greater than 1, what is the remainder when the product abc is divided by 2?

(Statement1): If each of a, b, and c is divided by 2, the product of all three remainders is 0.
--> At least one of a,b and c is EVEN. --> abc- EVEN --> remainder will be ZERO.
Sufficient

(Statement2): If each of a, b, and c is divided by 2, the sum of all three remainders is 2.
--> One of a,b and c must be EVEN number. --> abc-EVEN -- remainder will be ZERO.
Sufficient

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Re: If a, b, and c are three positive integers, each greater than 1, what  [#permalink]

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(2) If each of a, b, and c is divided by 2, the sum of all three remainders is 2.

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Re: If a, b, and c are three positive integers, each greater than 1, what  [#permalink]

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1
If at least one of a,b,c is even, the remainder of product abc, when divided by 2, is zero (0)

(1) If each of a, b, and c is divided by 2, the product of all three remainders is 0.
At least one of the three remainders is 0. Thus, at least one of a, b, and c is even and consequently, the remainder of product abc, when divided by 2, must be zero
SUFFICIENT

(2) If each of a, b, and c is divided by 2, the sum of all three remainders is 2.
the three remainders set can only be {0,1,1} and thus, at least one of the three remainders is 0. Therefore, at least one of a, b, and c is even and consequently, the remainder of product abc, when divided by 2, must be zero.
SUFFICIENT

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Re: If a, b, and c are three positive integers, each greater than 1, what  [#permalink]

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1
If a, b, and c are three positive integers, each greater than 1, what is the remainder when the product abc is divided by 2?

(1) If each of a, b, and c is divided by 2, the product of all three remainders is 0.

(2) If each of a, b, and c is divided by 2, the sum of all three remainders is 2.

Solution:
From statement (1), any one of a, b, c must be even. Sufficient.
From statement (2), any one of a, b, c must be even. Sufficient.

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Re: If a, b, and c are three positive integers, each greater than 1, what  [#permalink]

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what is the remainder when the product abc is divided by 2..............for the product to be divisible.....atleast one out of a,b,c must be divisible by 2

(1) If each of a, b, and c is divided by 2, the product of all three remainders is 0......this meant atleast one of the remainders is 0......so sufficient to say that the product is divisible by 2...........CORRECT

(2) If each of a, b, and c is divided by 2, the sum of all three remainders is 2........
for this the possiblity is 1,1,0 or 2,0,0....so in both the cases...atleast on eof them is zero....so sufficient to say that the product is divisible by 2...........CORRECT

OA:D Re: If a, b, and c are three positive integers, each greater than 1, what   [#permalink] 26 Feb 2020, 23:09
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