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There are three positive integers. The sum of the squares of the any two of these equals the sum of the product of these two numbers and the squares of the third number. How many of the three numbers must be equal?
- a)0
- b)2
- c)3
- d)Cannot be determined
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
There are three positive integers. The sum of the squares of the any t...
Let the numbers be a, b, c.
Now,
Now,
Adding these we get,
This is possible only when a - b = b - c = c - a = 0
∴ a = b = c
Hence, all three numbers must be equal.
Hence, option 3.
Hence, option 3.
Most Upvoted Answer
There are three positive integers. The sum of the squares of the any t...
Let's take the number 3
Therefore, 3^2 + 3^2 = 18 (LHS)
Also,
(3×3)+3^2 = 18 (RHS)
TRY IT OUT FOR DIFFERENT NUMBERS AS WELL.
Therefore, 3^2 + 3^2 = 18 (LHS)
Also,
(3×3)+3^2 = 18 (RHS)
TRY IT OUT FOR DIFFERENT NUMBERS AS WELL.
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Community Answer
There are three positive integers. The sum of the squares of the any t...
Solution:
Let's assume the three positive integers as a, b, and c.
Given that the sum of the squares of any two numbers is equal to the sum of the product of these two numbers and the square of the third number.
So, we can write the given condition as:
a^2 + b^2 = (a*b*c) + c^2 ...(1)
Similarly, we can write two more equations for the other two pairs of numbers:
b^2 + c^2 = (b*c*a) + a^2 ...(2)
c^2 + a^2 = (c*a*b) + b^2 ...(3)
Simplifying equations (1), (2), and (3), we get:
a^2 + b^2 - a*b*c = c^2 ...(4)
b^2 + c^2 - b*c*a = a^2 ...(5)
c^2 + a^2 - c*a*b = b^2 ...(6)
From equation (4), we can write:
a^2 - a*b*c = c^2 - b^2
Simplifying further, we get:
a^2 - b^2 = c^2 - a*b*c
Factoring both sides, we get:
(a - b)(a + b) = c(c - a*b) ...(7)
Similarly, from equations (5) and (6), we can write:
(b - c)(b + c) = a(a - b*c) ...(8)
(c - a)(c + a) = b(b - c*a) ...(9)
Now, let's analyze the options:
a) 0: If all three numbers are different, then none of the equations (7), (8), and (9) will hold true. Therefore, option (a) is not correct.
b) 2: If two of the numbers are equal, let's say a = b, then equation (7) becomes:
0 = c(c - a*b)
This implies c = 0 or c = a*b. Similarly, we can analyze equations (8) and (9). In both cases, we find that two of the numbers must be equal. Therefore, option (b) is a possible answer.
c) 3: If all three numbers are equal, then equations (7), (8), and (9) will hold true. Therefore, option (c) is a possible answer.
d) Cannot be determined: From our analysis, we can determine that the answer is either option (b) or option (c). Therefore, option (d) is not correct.
Hence, the correct answer is option (c) - 3.
Let's assume the three positive integers as a, b, and c.
Given that the sum of the squares of any two numbers is equal to the sum of the product of these two numbers and the square of the third number.
So, we can write the given condition as:
a^2 + b^2 = (a*b*c) + c^2 ...(1)
Similarly, we can write two more equations for the other two pairs of numbers:
b^2 + c^2 = (b*c*a) + a^2 ...(2)
c^2 + a^2 = (c*a*b) + b^2 ...(3)
Simplifying equations (1), (2), and (3), we get:
a^2 + b^2 - a*b*c = c^2 ...(4)
b^2 + c^2 - b*c*a = a^2 ...(5)
c^2 + a^2 - c*a*b = b^2 ...(6)
From equation (4), we can write:
a^2 - a*b*c = c^2 - b^2
Simplifying further, we get:
a^2 - b^2 = c^2 - a*b*c
Factoring both sides, we get:
(a - b)(a + b) = c(c - a*b) ...(7)
Similarly, from equations (5) and (6), we can write:
(b - c)(b + c) = a(a - b*c) ...(8)
(c - a)(c + a) = b(b - c*a) ...(9)
Now, let's analyze the options:
a) 0: If all three numbers are different, then none of the equations (7), (8), and (9) will hold true. Therefore, option (a) is not correct.
b) 2: If two of the numbers are equal, let's say a = b, then equation (7) becomes:
0 = c(c - a*b)
This implies c = 0 or c = a*b. Similarly, we can analyze equations (8) and (9). In both cases, we find that two of the numbers must be equal. Therefore, option (b) is a possible answer.
c) 3: If all three numbers are equal, then equations (7), (8), and (9) will hold true. Therefore, option (c) is a possible answer.
d) Cannot be determined: From our analysis, we can determine that the answer is either option (b) or option (c). Therefore, option (d) is not correct.
Hence, the correct answer is option (c) - 3.
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There are three positive integers. The sum of the squares of the any two of these equals the sum of the product of these two numbers and the squares of the third number. How many of the three numbers must be equal?a)0b)2c)3d)Cannot be determinedCorrect answer is option 'C'. Can you explain this answer?
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