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The sum of the factorials of the three-digits of a 3-digit number is equal to the three-digit number formed by these three digits, taken in the same order. Which of the following is true of the number of such three-digit numbers, if no digit occurs more than once?
- a)No such number exists
- b)Exactly one such number exists
- c)There is more than one such number, but they are finite in number
- d)There are infinite such numbers
Correct answer is 'B'. Can you explain this answer?
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The sum of the factorials of the three-digits of a 3-digit number is e...
There is only one number, 145, which exhibits this property.
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The sum of the factorials of the three-digits of a 3-digit number is e...
Solution:
Let the three-digit number be represented by $abc$, where $a$, $b$, and $c$ are digits.
The problem states that the sum of the factorials of the digits of the number is equal to the number itself. Therefore, we can write:
$abc = a! + b! + c!$
Observations:
1. The maximum value of $a!$ is $6! = 720$, which is less than $1000$. Therefore, $a$ can be at most $6$.
2. If $a = 5$ or $a = 6$, then $a!$ is a multiple of $5$ and $b!$ and $c!$ are at most $4! = 24$. Therefore, the sum $a! + b! + c!$ is less than $120 + 24 + 24 = 168$, which is less than $500$. Therefore, $a$ can be at most $4$.
3. If $a = 4$, then $a!$ is a multiple of $24$ and $b!$ and $c!$ are at most $3! = 6$. Therefore, the sum $a! + b! + c!$ is less than $24 \cdot 4 + 6 + 6 = 102$, which is less than $400$. Therefore, $a$ can be at most $3$.
Using these observations, we can now check all possible values of $a$, $b$, and $c$:
1. If $a = 1$, then $b! + c! = 100 - 1 = 99$. The only solution is $b = 4$ and $c = 5$, giving the number $145$, which is the only solution.
2. If $a = 2$, then $2b! + 2c! = 200 - 2 = 198$. This equation has no solutions, since $b!$ and $c!$ are both even.
3. If $a = 3$, then $6 + 6b! + 6c! = 300 + 3 - 6 = 297$. Dividing by $6$, we get $b! + c! = 49$. The only solutions are $b = 3$ and $c = 4$ or $b = 4$ and $c = 3$, giving the numbers $349$ and $439$.
Therefore, the only three-digit number that satisfies the condition is $145$. The answer is (B).
Let the three-digit number be represented by $abc$, where $a$, $b$, and $c$ are digits.
The problem states that the sum of the factorials of the digits of the number is equal to the number itself. Therefore, we can write:
$abc = a! + b! + c!$
Observations:
1. The maximum value of $a!$ is $6! = 720$, which is less than $1000$. Therefore, $a$ can be at most $6$.
2. If $a = 5$ or $a = 6$, then $a!$ is a multiple of $5$ and $b!$ and $c!$ are at most $4! = 24$. Therefore, the sum $a! + b! + c!$ is less than $120 + 24 + 24 = 168$, which is less than $500$. Therefore, $a$ can be at most $4$.
3. If $a = 4$, then $a!$ is a multiple of $24$ and $b!$ and $c!$ are at most $3! = 6$. Therefore, the sum $a! + b! + c!$ is less than $24 \cdot 4 + 6 + 6 = 102$, which is less than $400$. Therefore, $a$ can be at most $3$.
Using these observations, we can now check all possible values of $a$, $b$, and $c$:
1. If $a = 1$, then $b! + c! = 100 - 1 = 99$. The only solution is $b = 4$ and $c = 5$, giving the number $145$, which is the only solution.
2. If $a = 2$, then $2b! + 2c! = 200 - 2 = 198$. This equation has no solutions, since $b!$ and $c!$ are both even.
3. If $a = 3$, then $6 + 6b! + 6c! = 300 + 3 - 6 = 297$. Dividing by $6$, we get $b! + c! = 49$. The only solutions are $b = 3$ and $c = 4$ or $b = 4$ and $c = 3$, giving the numbers $349$ and $439$.
Therefore, the only three-digit number that satisfies the condition is $145$. The answer is (B).
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The sum of the factorials of the three-digits of a 3-digit number is equal to the three-digit number formed by these three digits, taken in the same order. Which of the following is true of the number of such three-digit numbers, if no digit occurs more than once?a)No such number existsb)Exactly one such number existsc)There is more than one such number, but they are finite in numberd)There are infinite such numbersCorrect answer is 'B'. Can you explain this answer?
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The sum of the factorials of the three-digits of a 3-digit number is equal to the three-digit number formed by these three digits, taken in the same order. Which of the following is true of the number of such three-digit numbers, if no digit occurs more than once?a)No such number existsb)Exactly one such number existsc)There is more than one such number, but they are finite in numberd)There are infinite such numbersCorrect answer is 'B'. Can you explain this answer? for Quant 2024 is part of Quant preparation. The Question and answers have been prepared according to the Quant exam syllabus. Information about The sum of the factorials of the three-digits of a 3-digit number is equal to the three-digit number formed by these three digits, taken in the same order. Which of the following is true of the number of such three-digit numbers, if no digit occurs more than once?a)No such number existsb)Exactly one such number existsc)There is more than one such number, but they are finite in numberd)There are infinite such numbersCorrect answer is 'B'. Can you explain this answer? covers all topics & solutions for Quant 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The sum of the factorials of the three-digits of a 3-digit number is equal to the three-digit number formed by these three digits, taken in the same order. Which of the following is true of the number of such three-digit numbers, if no digit occurs more than once?a)No such number existsb)Exactly one such number existsc)There is more than one such number, but they are finite in numberd)There are infinite such numbersCorrect answer is 'B'. Can you explain this answer?.
The sum of the factorials of the three-digits of a 3-digit number is equal to the three-digit number formed by these three digits, taken in the same order. Which of the following is true of the number of such three-digit numbers, if no digit occurs more than once?a)No such number existsb)Exactly one such number existsc)There is more than one such number, but they are finite in numberd)There are infinite such numbersCorrect answer is 'B'. Can you explain this answer? for Quant 2024 is part of Quant preparation. The Question and answers have been prepared according to the Quant exam syllabus. Information about The sum of the factorials of the three-digits of a 3-digit number is equal to the three-digit number formed by these three digits, taken in the same order. Which of the following is true of the number of such three-digit numbers, if no digit occurs more than once?a)No such number existsb)Exactly one such number existsc)There is more than one such number, but they are finite in numberd)There are infinite such numbersCorrect answer is 'B'. Can you explain this answer? covers all topics & solutions for Quant 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The sum of the factorials of the three-digits of a 3-digit number is equal to the three-digit number formed by these three digits, taken in the same order. Which of the following is true of the number of such three-digit numbers, if no digit occurs more than once?a)No such number existsb)Exactly one such number existsc)There is more than one such number, but they are finite in numberd)There are infinite such numbersCorrect answer is 'B'. Can you explain this answer?.
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