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A teacher wrote a number on the blackboard and the following observations were made by the students The number is a four-digit number.The sum of the digits equals the product of the digits. The number is divisible by the sum of the digits.The sum of the digits of the number is
- a)8
- b)10
- c)12
- d)14
Correct answer is 'A'. Can you explain this answer?
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A teacher wrote a number on the blackboard and the following observati...
Using options, the only possible value is 4112. The key here is: The sum of the digits equals the product of the digits.
Most Upvoted Answer
A teacher wrote a number on the blackboard and the following observati...
**Explanation:**
To solve this problem, let's consider the given conditions:
1. The sum of the digits equals the product of the digits.
2. The number is divisible by the sum of the digits.
Let's assume the number as ABCD, where A, B, C, and D represent the thousands, hundreds, tens, and units digits respectively.
**Condition 1: The sum of the digits equals the product of the digits.**
This condition implies that A + B + C + D = A * B * C * D.
We can consider the following cases:
**Case 1: A = 1**
If A = 1, then A * B * C * D = B * C * D. But the sum of the digits is A + B + C + D = 1 + B + C + D, which is not equal to B * C * D. Therefore, A cannot be 1.
**Case 2: A = 2**
If A = 2, then A * B * C * D = 2 * B * C * D. But the sum of the digits is A + B + C + D = 2 + B + C + D, which is not equal to 2 * B * C * D. Therefore, A cannot be 2.
**Case 3: A = 3**
If A = 3, then A * B * C * D = 3 * B * C * D. But the sum of the digits is A + B + C + D = 3 + B + C + D, which is not equal to 3 * B * C * D. Therefore, A cannot be 3.
**Case 4: A = 4**
If A = 4, then A * B * C * D = 4 * B * C * D. The sum of the digits is A + B + C + D = 4 + B + C + D, which is equal to 4 * B * C * D.
Therefore, the only possible value for A is 4.
**Condition 2: The number is divisible by the sum of the digits.**
Since A = 4, the number ABCD is divisible by the sum of the digits (A + B + C + D). Therefore, A + B + C + D must be a divisor of 4.
The divisors of 4 are 1, 2, and 4.
Let's consider each case:
1. If A + B + C + D = 1, then it is not possible as the sum of the digits cannot be 1 for a four-digit number.
2. If A + B + C + D = 2, then it is not possible as the sum of the digits cannot be 2 for a four-digit number.
3. If A + B + C + D = 4, then it is possible as the sum of the digits can be 4 for a four-digit number.
Therefore, the sum of the digits of the number is 4.
Hence, the correct answer is option **A) 8**.
To solve this problem, let's consider the given conditions:
1. The sum of the digits equals the product of the digits.
2. The number is divisible by the sum of the digits.
Let's assume the number as ABCD, where A, B, C, and D represent the thousands, hundreds, tens, and units digits respectively.
**Condition 1: The sum of the digits equals the product of the digits.**
This condition implies that A + B + C + D = A * B * C * D.
We can consider the following cases:
**Case 1: A = 1**
If A = 1, then A * B * C * D = B * C * D. But the sum of the digits is A + B + C + D = 1 + B + C + D, which is not equal to B * C * D. Therefore, A cannot be 1.
**Case 2: A = 2**
If A = 2, then A * B * C * D = 2 * B * C * D. But the sum of the digits is A + B + C + D = 2 + B + C + D, which is not equal to 2 * B * C * D. Therefore, A cannot be 2.
**Case 3: A = 3**
If A = 3, then A * B * C * D = 3 * B * C * D. But the sum of the digits is A + B + C + D = 3 + B + C + D, which is not equal to 3 * B * C * D. Therefore, A cannot be 3.
**Case 4: A = 4**
If A = 4, then A * B * C * D = 4 * B * C * D. The sum of the digits is A + B + C + D = 4 + B + C + D, which is equal to 4 * B * C * D.
Therefore, the only possible value for A is 4.
**Condition 2: The number is divisible by the sum of the digits.**
Since A = 4, the number ABCD is divisible by the sum of the digits (A + B + C + D). Therefore, A + B + C + D must be a divisor of 4.
The divisors of 4 are 1, 2, and 4.
Let's consider each case:
1. If A + B + C + D = 1, then it is not possible as the sum of the digits cannot be 1 for a four-digit number.
2. If A + B + C + D = 2, then it is not possible as the sum of the digits cannot be 2 for a four-digit number.
3. If A + B + C + D = 4, then it is possible as the sum of the digits can be 4 for a four-digit number.
Therefore, the sum of the digits of the number is 4.
Hence, the correct answer is option **A) 8**.
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A teacher wrote a number on the blackboard and the following observations were made by the students The number is a four-digit number.The sum of the digits equals the product of the digits. The number is divisible by the sum of the digits.The sum of the digits of the number isa)8b)10c)12d)14Correct answer is 'A'. Can you explain this answer?
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