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If x represents the sum of all the positive three-digit numbers that can be constructed using each of the
distinct nonzero digits a, b, and c exactly once, what is the largest integer by which x must be divisible?
distinct nonzero digits a, b, and c exactly once, what is the largest integer by which x must be divisible?
- a)3
- b)6
- c)11
- d)22
- e)222
Correct answer is option 'E'. Can you explain this answer?
Verified Answer
If x represents the sum of all the positive three-digit numbers that c...
There are 3!, or 6, different three-digit numbers that can be constructed using the digits a, b, and c:
The value of any one of these numbers can be represented using place values. For example, the value of abc is 100a + 10b + c.
Therefore, you can represent the sum of the 6 numbers as:
=222(a+b+c)
x is equal to 222(a + b + c). Therefore, x must be divisible by 222.
The correct answer is E
The value of any one of these numbers can be represented using place values. For example, the value of abc is 100a + 10b + c.
Therefore, you can represent the sum of the 6 numbers as:
=222(a+b+c)
x is equal to 222(a + b + c). Therefore, x must be divisible by 222.
The correct answer is E
Most Upvoted Answer
If x represents the sum of all the positive three-digit numbers that c...
Answer:
To find the sum of all the positive three-digit numbers that can be constructed using distinct nonzero digits a, b, and c exactly once, we can consider the possible values of each digit in the hundreds, tens, and units place.
Step 1: Determine the possible values for each digit:
Since the digits are distinct nonzero digits, we have the following possibilities:
Hundreds place: a, b, c (3 possibilities)
Tens place: a, b, c (3 possibilities)
Units place: a, b, c (3 possibilities)
Step 2: Calculate the sum of all possible three-digit numbers:
To find the sum, we need to consider all possible combinations of the digits in each place value.
For the hundreds place, the sum of all possible digits is (a + b + c).
For the tens place, the sum of all possible digits is (a + b + c).
For the units place, the sum of all possible digits is (a + b + c).
So, the sum of all possible three-digit numbers is (a + b + c) * 111.
Step 3: Determine the largest integer by which the sum must be divisible:
To find the largest integer by which the sum must be divisible, we need to find the greatest common divisor (GCD) of (a + b + c) and 111.
The GCD of (a + b + c) and 111 will be the largest integer by which the sum must be divisible.
Step 4: Finding the GCD:
To find the GCD, we can use the Euclidean algorithm.
Let's assume a > b > c. Since the digits are distinct, the values of a, b, and c can be represented as a = 9, b = 8, and c = 7.
(a + b + c) = (9 + 8 + 7) = 24.
Now, we find the GCD of 24 and 111 using the Euclidean algorithm:
111 = 24 * 4 + 15
24 = 15 * 1 + 9
15 = 9 * 1 + 6
9 = 6 * 1 + 3
6 = 3 * 2 + 0
Since the remainder is 0, the GCD is 3.
Step 5: Final Answer:
Therefore, the largest integer by which the sum must be divisible is 3. However, the correct answer option given is 222, which seems to be incorrect based on the calculations provided.
To find the sum of all the positive three-digit numbers that can be constructed using distinct nonzero digits a, b, and c exactly once, we can consider the possible values of each digit in the hundreds, tens, and units place.
Step 1: Determine the possible values for each digit:
Since the digits are distinct nonzero digits, we have the following possibilities:
Hundreds place: a, b, c (3 possibilities)
Tens place: a, b, c (3 possibilities)
Units place: a, b, c (3 possibilities)
Step 2: Calculate the sum of all possible three-digit numbers:
To find the sum, we need to consider all possible combinations of the digits in each place value.
For the hundreds place, the sum of all possible digits is (a + b + c).
For the tens place, the sum of all possible digits is (a + b + c).
For the units place, the sum of all possible digits is (a + b + c).
So, the sum of all possible three-digit numbers is (a + b + c) * 111.
Step 3: Determine the largest integer by which the sum must be divisible:
To find the largest integer by which the sum must be divisible, we need to find the greatest common divisor (GCD) of (a + b + c) and 111.
The GCD of (a + b + c) and 111 will be the largest integer by which the sum must be divisible.
Step 4: Finding the GCD:
To find the GCD, we can use the Euclidean algorithm.
Let's assume a > b > c. Since the digits are distinct, the values of a, b, and c can be represented as a = 9, b = 8, and c = 7.
(a + b + c) = (9 + 8 + 7) = 24.
Now, we find the GCD of 24 and 111 using the Euclidean algorithm:
111 = 24 * 4 + 15
24 = 15 * 1 + 9
15 = 9 * 1 + 6
9 = 6 * 1 + 3
6 = 3 * 2 + 0
Since the remainder is 0, the GCD is 3.
Step 5: Final Answer:
Therefore, the largest integer by which the sum must be divisible is 3. However, the correct answer option given is 222, which seems to be incorrect based on the calculations provided.
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If x represents the sum of all the positive three-digit numbers that can be constructed using each of thedistinct nonzero digits a, b, and c exactly once, what is the largest integer by which x must be divisible?a)3b)6c)11d)22e)222Correct answer is option 'E'. Can you explain this answer? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If x represents the sum of all the positive three-digit numbers that can be constructed using each of thedistinct nonzero digits a, b, and c exactly once, what is the largest integer by which x must be divisible?a)3b)6c)11d)22e)222Correct answer is option 'E'. Can you explain this answer? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If x represents the sum of all the positive three-digit numbers that can be constructed using each of thedistinct nonzero digits a, b, and c exactly once, what is the largest integer by which x must be divisible?a)3b)6c)11d)22e)222Correct answer is option 'E'. Can you explain this answer?.
If x represents the sum of all the positive three-digit numbers that can be constructed using each of thedistinct nonzero digits a, b, and c exactly once, what is the largest integer by which x must be divisible?a)3b)6c)11d)22e)222Correct answer is option 'E'. Can you explain this answer? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If x represents the sum of all the positive three-digit numbers that can be constructed using each of thedistinct nonzero digits a, b, and c exactly once, what is the largest integer by which x must be divisible?a)3b)6c)11d)22e)222Correct answer is option 'E'. Can you explain this answer? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If x represents the sum of all the positive three-digit numbers that can be constructed using each of thedistinct nonzero digits a, b, and c exactly once, what is the largest integer by which x must be divisible?a)3b)6c)11d)22e)222Correct answer is option 'E'. Can you explain this answer?.
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