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Let S be a set of first 10 natural numbers. What is the possible number of pairs (a, b) where a, b ∈ S and a ≠ b such that the product ab (>12) leaves remainder 4 when divided by 12?
- a)4
- b)6
- c)8
- d)10
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Let S be a set of first 10 natural numbers. What is the possible numbe...
S is a set of first 10 natural numbers;
∴ S = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10)
∵ a, b ∈ S and a ≠ b such that ab > 12;
∴ S = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10)
∵ a, b ∈ S and a ≠ b such that ab > 12;
∴ The product ab can’t be any 3 digit number (it will be 2 digit number)
∴ Product ab (>12) that leaves remainder 4 when divided by 12 = (12n + 4);
∴ All such number (2 digit only) will be 16, 28, 40, 52, 64, 76, 88
Now classifying these numbers as the product ab in the pair (a, b) where a, b ∈ S a ≠ b:
16: (2, 8), (8, 2)
28: (4, 7), (7, 4)
40: (5, 8), (8, 5), (4, 10), (10, 4)
52: None
64: None
76: None
88: None
∴ Number of total pairs = 8
28: (4, 7), (7, 4)
40: (5, 8), (8, 5), (4, 10), (10, 4)
52: None
64: None
76: None
88: None
∴ Number of total pairs = 8
Most Upvoted Answer
Let S be a set of first 10 natural numbers. What is the possible numbe...
Explanation:
The remainders when a number is divided by 12 can be 0, 1, 2, ..., 11. Since the product of a and b leaves a remainder of 4 when divided by 12, we can express a = 12m + 4 and b = 12n + 4, where m and n are integers.
Finding Possible Pairs:
- Substituting the values of a and b in ab > 12, we get (12m + 4)(12n + 4) > 12.
- Expanding the expression, we get 144mn + 48m + 48n + 16 > 12.
- Simplifying, we get 144mn + 48m + 48n + 4 > 0.
- Dividing by 4, we get 36mn + 12m + 12n + 1 > 0.
- Rearranging terms, we get (6m + 1)(6n + 1) > 0.
Possible Values:
- The inequality (6m + 1)(6n + 1) > 0 holds true for all values of m and n except when both are negative.
- Since we are dealing with the first 10 natural numbers, m and n can vary from 0 to 9.
- Thus, the possible number of pairs (a, b) is 8, as there are 8 pairs that satisfy the given conditions.
Therefore, the correct answer is option 'C' (8).
The remainders when a number is divided by 12 can be 0, 1, 2, ..., 11. Since the product of a and b leaves a remainder of 4 when divided by 12, we can express a = 12m + 4 and b = 12n + 4, where m and n are integers.
Finding Possible Pairs:
- Substituting the values of a and b in ab > 12, we get (12m + 4)(12n + 4) > 12.
- Expanding the expression, we get 144mn + 48m + 48n + 16 > 12.
- Simplifying, we get 144mn + 48m + 48n + 4 > 0.
- Dividing by 4, we get 36mn + 12m + 12n + 1 > 0.
- Rearranging terms, we get (6m + 1)(6n + 1) > 0.
Possible Values:
- The inequality (6m + 1)(6n + 1) > 0 holds true for all values of m and n except when both are negative.
- Since we are dealing with the first 10 natural numbers, m and n can vary from 0 to 9.
- Thus, the possible number of pairs (a, b) is 8, as there are 8 pairs that satisfy the given conditions.
Therefore, the correct answer is option 'C' (8).
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Let S be a set of first 10 natural numbers. What is the possible number of pairs (a, b) where a, b ∈ S and a ≠ b such that the product ab (>12) leaves remainder 4 when divided by 12?a)4b)6c)8d)10Correct answer is option 'C'. Can you explain this answer?
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Let S be a set of first 10 natural numbers. What is the possible number of pairs (a, b) where a, b ∈ S and a ≠ b such that the product ab (>12) leaves remainder 4 when divided by 12?a)4b)6c)8d)10Correct answer is option 'C'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about Let S be a set of first 10 natural numbers. What is the possible number of pairs (a, b) where a, b ∈ S and a ≠ b such that the product ab (>12) leaves remainder 4 when divided by 12?a)4b)6c)8d)10Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let S be a set of first 10 natural numbers. What is the possible number of pairs (a, b) where a, b ∈ S and a ≠ b such that the product ab (>12) leaves remainder 4 when divided by 12?a)4b)6c)8d)10Correct answer is option 'C'. Can you explain this answer?.
Let S be a set of first 10 natural numbers. What is the possible number of pairs (a, b) where a, b ∈ S and a ≠ b such that the product ab (>12) leaves remainder 4 when divided by 12?a)4b)6c)8d)10Correct answer is option 'C'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about Let S be a set of first 10 natural numbers. What is the possible number of pairs (a, b) where a, b ∈ S and a ≠ b such that the product ab (>12) leaves remainder 4 when divided by 12?a)4b)6c)8d)10Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let S be a set of first 10 natural numbers. What is the possible number of pairs (a, b) where a, b ∈ S and a ≠ b such that the product ab (>12) leaves remainder 4 when divided by 12?a)4b)6c)8d)10Correct answer is option 'C'. Can you explain this answer?.
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