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a, b, c are three distinct integers from 2 to 10 (both inclusive). Exactly one of ab, bc and ca is odd. abc is a multiple of 4. The arithmetic mean of a and b is an integer and so is the arithmetic mean of a, b and c. How many such triplets are possible (unordered triplets)?
- a)5
- b)3
- c)2
- d)4
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
a, b, c are three distinct integers from 2 to 10 (both inclusive). Ex...
Given Information:
- We have three distinct integers a, b, and c, which are between 2 and 10 (inclusive).
- Exactly one of the products ab, bc, and ca is odd.
- The product abc is a multiple of 4.
- The arithmetic mean of a and b is an integer, as well as the arithmetic mean of a, b, and c.
Approach:
To solve this problem, we need to consider the given conditions one by one and find the possible values of a, b, and c.
Condition 1: Exactly one of ab, bc, and ca is odd.
- For this condition to be true, one of the three pairs (a, b), (b, c), or (c, a) must be odd, while the other two pairs must be even.
- Since all the integers are distinct, we can have three possible cases:
1) a and b are odd, c is even
2) b and c are odd, a is even
3) c and a are odd, b is even
Condition 2: The product abc is a multiple of 4.
- For abc to be a multiple of 4, at least one of the three integers a, b, or c must be divisible by 4.
- We can have the following cases:
1) a is divisible by 4
2) b is divisible by 4
3) c is divisible by 4
Condition 3: The arithmetic mean of a and b is an integer, as well as the arithmetic mean of a, b, and c.
- The arithmetic mean of two integers is an integer if their sum is divisible by 2.
- The arithmetic mean of three integers is an integer if their sum is divisible by 3.
- We can have the following cases:
1) (a + b) is divisible by 2, (a + b + c) is divisible by 3
2) (b + c) is divisible by 2, (a + b + c) is divisible by 3
3) (c + a) is divisible by 2, (a + b + c) is divisible by 3
Solution:
By considering all the possible cases for each condition, we can find the values of a, b, and c that satisfy all the given conditions. Let's analyze each case:
Case 1: a and b are odd, c is even
- Possible values: (3, 5, 4), (3, 7, 4), (5, 7, 4)
Case 2: b and c are odd, a is even
- Possible values: (2, 3, 7), (2, 5, 7), (2, 7, 9), (2, 7, 5), (2, 7, 3)
Case 3: c and a are odd, b is even
- Possible values: (3, 4, 9), (3, 4, 7), (5, 4, 9), (5,
- We have three distinct integers a, b, and c, which are between 2 and 10 (inclusive).
- Exactly one of the products ab, bc, and ca is odd.
- The product abc is a multiple of 4.
- The arithmetic mean of a and b is an integer, as well as the arithmetic mean of a, b, and c.
Approach:
To solve this problem, we need to consider the given conditions one by one and find the possible values of a, b, and c.
Condition 1: Exactly one of ab, bc, and ca is odd.
- For this condition to be true, one of the three pairs (a, b), (b, c), or (c, a) must be odd, while the other two pairs must be even.
- Since all the integers are distinct, we can have three possible cases:
1) a and b are odd, c is even
2) b and c are odd, a is even
3) c and a are odd, b is even
Condition 2: The product abc is a multiple of 4.
- For abc to be a multiple of 4, at least one of the three integers a, b, or c must be divisible by 4.
- We can have the following cases:
1) a is divisible by 4
2) b is divisible by 4
3) c is divisible by 4
Condition 3: The arithmetic mean of a and b is an integer, as well as the arithmetic mean of a, b, and c.
- The arithmetic mean of two integers is an integer if their sum is divisible by 2.
- The arithmetic mean of three integers is an integer if their sum is divisible by 3.
- We can have the following cases:
1) (a + b) is divisible by 2, (a + b + c) is divisible by 3
2) (b + c) is divisible by 2, (a + b + c) is divisible by 3
3) (c + a) is divisible by 2, (a + b + c) is divisible by 3
Solution:
By considering all the possible cases for each condition, we can find the values of a, b, and c that satisfy all the given conditions. Let's analyze each case:
Case 1: a and b are odd, c is even
- Possible values: (3, 5, 4), (3, 7, 4), (5, 7, 4)
Case 2: b and c are odd, a is even
- Possible values: (2, 3, 7), (2, 5, 7), (2, 7, 9), (2, 7, 5), (2, 7, 3)
Case 3: c and a are odd, b is even
- Possible values: (3, 4, 9), (3, 4, 7), (5, 4, 9), (5,
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Community Answer
a, b, c are three distinct integers from 2 to 10 (both inclusive). Ex...
Exactly one of ab, bc and ca is odd ⇒ Two are odd and one is even.
abc is a multiple of 4 ⇒ the even number is a multiple of 4.
The arithmetic mean of a and b is an integer ⇒ a and b are odd.
and so is the arithmetic mean of a, b and c. ⇒ a + b + c is a multiple of 3.
c can be 4 or 8.
c = 4; a, b can be 3, 5 or 5, 9
c = 8; a, b can be 3, 7 or 7, 9
Four triplets are possible.
Hence, the correct option is (d).
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a, b, c are three distinct integers from 2 to 10 (both inclusive). Exactly one of ab, bc and ca is odd. abc is a multiple of 4. The arithmetic mean of a and b is an integer and so is the arithmetic mean of a, b and c. How many such triplets are possible (unordered triplets)?a)5b)3c)2d)4Correct answer is option 'D'. Can you explain this answer?
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