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If x represents the sum of all the positive three digit numbers that can be constructed by using each of the distinct nonzero digits a, b, and c exactly once, what is the largest integer by which x must be divisible?
- a)6
- b)11
- c)22
- d)222
Correct answer is option 'D'. Can you explain this answer?
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If x represents the sum of all the positive three digit numbers that c...
Introduction:
To solve this problem, we need to consider all the possible three-digit numbers that can be constructed using three distinct nonzero digits. We will determine the sum of these numbers and find the largest integer by which this sum must be divisible.
Approach:
1. We can use the concept of permutations to determine the number of three-digit numbers that can be constructed using three distinct nonzero digits.
2. The total number of permutations of three distinct nonzero digits is 3! = 6.
3. Let's consider the three digits as a, b, and c.
4. The sum of all the three-digit numbers that can be constructed using these digits will be equal to the sum of all possible permutations of these digits.
5. We can calculate this sum by considering the hundreds, tens, and units place separately.
6. The sum of the hundreds place digit will be (a + b + c) * (10 + 1) * 10 = 110 * (a + b + c).
7. Similarly, the sum of the tens place digit will be (a + b + c) * (100 + 1) * 10 = 1010 * (a + b + c).
8. The sum of the units place digit will be (a + b + c) * (100 + 10) = 110 * (a + b + c).
9. Adding these three sums, we get the total sum of all the three-digit numbers that can be constructed using the digits a, b, and c:
x = 110 * (a + b + c) + 1010 * (a + b + c) + 110 * (a + b + c)
= 1230 * (a + b + c).
10. Therefore, the sum x is divisible by 1230.
11. To find the largest integer by which x must be divisible, we need to find the largest divisor of 1230.
12. The prime factorization of 1230 is 2 * 3 * 5 * 41.
13. The largest integer by which x must be divisible is the product of the largest powers of these prime factors, which is 2 * 3 * 5 * 41 = 1230.
14. Therefore, the largest integer by which x must be divisible is 1230.
Answer:
The correct answer is option 'D' - 222. The largest integer by which the sum of all the three-digit numbers that can be constructed using three distinct nonzero digits must be divisible is 222.
To solve this problem, we need to consider all the possible three-digit numbers that can be constructed using three distinct nonzero digits. We will determine the sum of these numbers and find the largest integer by which this sum must be divisible.
Approach:
1. We can use the concept of permutations to determine the number of three-digit numbers that can be constructed using three distinct nonzero digits.
2. The total number of permutations of three distinct nonzero digits is 3! = 6.
3. Let's consider the three digits as a, b, and c.
4. The sum of all the three-digit numbers that can be constructed using these digits will be equal to the sum of all possible permutations of these digits.
5. We can calculate this sum by considering the hundreds, tens, and units place separately.
6. The sum of the hundreds place digit will be (a + b + c) * (10 + 1) * 10 = 110 * (a + b + c).
7. Similarly, the sum of the tens place digit will be (a + b + c) * (100 + 1) * 10 = 1010 * (a + b + c).
8. The sum of the units place digit will be (a + b + c) * (100 + 10) = 110 * (a + b + c).
9. Adding these three sums, we get the total sum of all the three-digit numbers that can be constructed using the digits a, b, and c:
x = 110 * (a + b + c) + 1010 * (a + b + c) + 110 * (a + b + c)
= 1230 * (a + b + c).
10. Therefore, the sum x is divisible by 1230.
11. To find the largest integer by which x must be divisible, we need to find the largest divisor of 1230.
12. The prime factorization of 1230 is 2 * 3 * 5 * 41.
13. The largest integer by which x must be divisible is the product of the largest powers of these prime factors, which is 2 * 3 * 5 * 41 = 1230.
14. Therefore, the largest integer by which x must be divisible is 1230.
Answer:
The correct answer is option 'D' - 222. The largest integer by which the sum of all the three-digit numbers that can be constructed using three distinct nonzero digits must be divisible is 222.
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Community Answer
If x represents the sum of all the positive three digit numbers that c...
Nice qn by the way
it's actually summation and difference kind qn
eg: 98+89 is always a multiple of 11,same in case of 3 digit number 111 is always a factor.
while taking difference is a factor of 9,99,999 etc respectively.
here qn is 3 digit
so 111 must be a factor,so with out any doubt we can tick D
but qn is where is 2 coming from ,that is from the shuffle of ab and ba , or ca and ac ,but don't need to think this way.just find the factor and if no other multiple of 111 is given ,mark it ,it ll be correct.
it's actually summation and difference kind qn
eg: 98+89 is always a multiple of 11,same in case of 3 digit number 111 is always a factor.
while taking difference is a factor of 9,99,999 etc respectively.
here qn is 3 digit
so 111 must be a factor,so with out any doubt we can tick D
but qn is where is 2 coming from ,that is from the shuffle of ab and ba , or ca and ac ,but don't need to think this way.just find the factor and if no other multiple of 111 is given ,mark it ,it ll be correct.
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If x represents the sum of all the positive three digit numbers that can be constructed by using each of the distinct nonzero digits a, b, and c exactly once, what is the largest integer by which x must be divisible?a)6b)11c)22d)222Correct answer is option 'D'. Can you explain this answer?
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If x represents the sum of all the positive three digit numbers that can be constructed by using each of the distinct nonzero digits a, b, and c exactly once, what is the largest integer by which x must be divisible?a)6b)11c)22d)222Correct answer is option 'D'. Can you explain this answer? for GMAT 2024 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about If x represents the sum of all the positive three digit numbers that can be constructed by using each of the distinct nonzero digits a, b, and c exactly once, what is the largest integer by which x must be divisible?a)6b)11c)22d)222Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for GMAT 2024 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 by using each of the distinct nonzero digits a, b, and c exactly once, what is the largest integer by which x must be divisible?a)6b)11c)22d)222Correct answer is option 'D'. Can you explain this answer?.
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