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01 Jan 2024, 02:30
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We define “a mod n” as the remainder when a positive integer a is divided by a positive integer n. What is the value of “a mod 6”?
(1) a mod 4 = a mod 12
(2) a mod 3 = 2
(1) a mod 4 = a mod 12
(2) a mod 3 = 2
01 Jan 2024, 05:00
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To determine the value of "a mod 6," we need to analyze the given statements to see if they provide sufficient information either individually or combined.
Statement 1: a mod 4 = a mod 12
This statement means that the remainder when a is divided by 4 is the same as the remainder when a is divided by 12.
This statement will hold true for multiple numbers such as 12, 13, 14 etc. Thus, we cannot determine the value of a mod 6 from Statement 1 alone.
Statement 2: a mod 3 = 2
This statement means that when a is divided by 3, the remainder is 2. For example, this would be true for numbers like 5, 8, 11, etc.
However, knowing only this information does not allow us to determine the value of a mod 6, as there are multiple possibilities for a that satisfy this condition. Thus, we cannot determine the value of a mod 6 from Statement 2 alone.
Combining Statements 1 and 2:
By combining the two statements, we know that a leaves a remainder of 2 when divided by 3 and that the remainder when a is divided by 4 is the same as when it is divided by 12. This information can help us narrow down the possible values for a.
However, even with both statements combined, there can be multiple values of a that satisfy both conditions. For example, a = 14 satisfies both conditions (14 mod 3 = 2, and 14 mod 4 = 2 which is equal to 14 mod 12 = 2), and a = 26 also satisfies both conditions (26 mod 3 = 2, and 26 mod 4 = 2 which is equal to 26 mod 12 = 2).
But, 14 mod 6 = 2 and 26 mod 6 = 2.
In fact, when considering both statements, any number of the form 12k + 2 (where k is an integer) would satisfy both conditions and yield a mod 6 = 2.
Thus, by combining both statements, we can conclude that a mod 6 = 2.
Thus, both Statements together are sufficient.
Statement 1: a mod 4 = a mod 12
This statement means that the remainder when a is divided by 4 is the same as the remainder when a is divided by 12.
This statement will hold true for multiple numbers such as 12, 13, 14 etc. Thus, we cannot determine the value of a mod 6 from Statement 1 alone.
Statement 2: a mod 3 = 2
This statement means that when a is divided by 3, the remainder is 2. For example, this would be true for numbers like 5, 8, 11, etc.
However, knowing only this information does not allow us to determine the value of a mod 6, as there are multiple possibilities for a that satisfy this condition. Thus, we cannot determine the value of a mod 6 from Statement 2 alone.
Combining Statements 1 and 2:
By combining the two statements, we know that a leaves a remainder of 2 when divided by 3 and that the remainder when a is divided by 4 is the same as when it is divided by 12. This information can help us narrow down the possible values for a.
However, even with both statements combined, there can be multiple values of a that satisfy both conditions. For example, a = 14 satisfies both conditions (14 mod 3 = 2, and 14 mod 4 = 2 which is equal to 14 mod 12 = 2), and a = 26 also satisfies both conditions (26 mod 3 = 2, and 26 mod 4 = 2 which is equal to 26 mod 12 = 2).
But, 14 mod 6 = 2 and 26 mod 6 = 2.
In fact, when considering both statements, any number of the form 12k + 2 (where k is an integer) would satisfy both conditions and yield a mod 6 = 2.
Thus, by combining both statements, we can conclude that a mod 6 = 2.
Thus, both Statements together are sufficient.
01 Jan 2024, 06:36
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1) a mod 4 = a mod 12
The Remainder < 4 & < 12. Then R <4 {1,2,3}
2) a mod 3 = 2
It means 3n+2 which is {2,5,8,11,...}
1 & 2) Combine we know the integer in question is 2
Then we can determine a mod 6.
ANS C
I think I oversimplfy this question, please Experts guide us
The Remainder < 4 & < 12. Then R <4 {1,2,3}
2) a mod 3 = 2
It means 3n+2 which is {2,5,8,11,...}
1 & 2) Combine we know the integer in question is 2
Then we can determine a mod 6.
ANS C
I think I oversimplfy this question, please Experts guide us
