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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 option 'A'. Can you explain this answer?
Verified Answer
A teacher wrote a number on the blackboard and the following observati...
Let the four-digit number be abcd, where a, b, c, and d are the digits of the number from the thousands place to the ones place.
Since the sum of the digits equals the product of the digits, we have:
a + b + c + d = abcd
And since the number is divisible by the sum of the digits, we have:
abcd / (a + b + c + d) = integer
Combining the two equations, we get:
abcd / (abcd) = integer
1 = integer
This means that the sum of the digits of the number must be equal to 1, which is only possible if the digits are 1, 1, and two 0s. However, this can't be a four-digit number since the thousands place can't be 0.
We also know that the sum of the digits is a factor of the four-digit number. Let's try combinations of numbers that multiply to a four-digit number.
Since the number is divisible by the sum of the digits, we can try to find a sum of digits that is a factor of a four-digit number. For example, if we try a sum of 8, we can find a four-digit number that is divisible by 8.
Let's try a sum of 8:
a + b + c + d = 8
abcd / 8 = integer
We need the product of the digits to also be 8, so the possible sets of digits are:
1, 1, 2, 4 (product = 8)
1, 1, 1, 8 (product = 8)
We can eliminate the second option (1, 1, 1, 8) since the maximum product we can get from three 1s and an 8 is 8, which is not a four-digit number.
Now let's try the first option (1, 1, 2, 4). We can arrange these digits to form different four-digit numbers and check if any of them are divisible by 8:
1124: not divisible by 8
1142: not divisible by 8
1214: not divisible by 8
1241: not divisible by 8
1412: not divisible by 8
1421: not divisible by 8
2114: not divisible by 8
2141: not divisible by 8
2411: not divisible by 8
4112: divisible by 8
4121: not divisible by 8
The number 4112 is divisible by 8 and has a sum of digits equal to 8 (1 + 1 + 2 + 4 = 8). Therefore, the sum of the digits of the number is 8.
Since the sum of the digits equals the product of the digits, we have:
a + b + c + d = abcd
And since the number is divisible by the sum of the digits, we have:
abcd / (a + b + c + d) = integer
Combining the two equations, we get:
abcd / (abcd) = integer
1 = integer
This means that the sum of the digits of the number must be equal to 1, which is only possible if the digits are 1, 1, and two 0s. However, this can't be a four-digit number since the thousands place can't be 0.
We also know that the sum of the digits is a factor of the four-digit number. Let's try combinations of numbers that multiply to a four-digit number.
Since the number is divisible by the sum of the digits, we can try to find a sum of digits that is a factor of a four-digit number. For example, if we try a sum of 8, we can find a four-digit number that is divisible by 8.
Let's try a sum of 8:
a + b + c + d = 8
abcd / 8 = integer
We need the product of the digits to also be 8, so the possible sets of digits are:
1, 1, 2, 4 (product = 8)
1, 1, 1, 8 (product = 8)
We can eliminate the second option (1, 1, 1, 8) since the maximum product we can get from three 1s and an 8 is 8, which is not a four-digit number.
Now let's try the first option (1, 1, 2, 4). We can arrange these digits to form different four-digit numbers and check if any of them are divisible by 8:
1124: not divisible by 8
1142: not divisible by 8
1214: not divisible by 8
1241: not divisible by 8
1412: not divisible by 8
1421: not divisible by 8
2114: not divisible by 8
2141: not divisible by 8
2411: not divisible by 8
4112: divisible by 8
4121: not divisible by 8
The number 4112 is divisible by 8 and has a sum of digits equal to 8 (1 + 1 + 2 + 4 = 8). Therefore, the sum of the digits of the number is 8.
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Most Upvoted Answer
A teacher wrote a number on the blackboard and the following observati...
Given information:
- 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.
Let's use the process of elimination to find the correct answer.
Option B (10): The only four-digit number whose digits add up to 10 is 1189. However, the product of its digits is 1 x 1 x 8 x 9 = 72, which is not equal to 10. Therefore, option B is incorrect.
Option C (12): The only four-digit number whose digits add up to 12 is 1233. However, the product of its digits is 1 x 2 x 3 x 3 = 18, which is not equal to 12. Therefore, option C is incorrect.
Option D (14): The only four-digit number whose digits add up to 14 is 1124. However, the product of its digits is 1 x 1 x 2 x 4 = 8, which is not equal to 14. Therefore, option D is incorrect.
Option A (8): The only four-digit number whose digits add up to 8 is 1124. The product of its digits is 1 x 1 x 2 x 4 = 8, which is equal to the sum of the digits. This satisfies the second condition. Also, the sum of the digits (1 + 1 + 2 + 4 = 8) divides the number (1124) completely, satisfying the third condition. Therefore, the correct answer is option A.
Hence, we can conclude that the sum of the digits of the number is 8.
- 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.
Let's use the process of elimination to find the correct answer.
Option B (10): The only four-digit number whose digits add up to 10 is 1189. However, the product of its digits is 1 x 1 x 8 x 9 = 72, which is not equal to 10. Therefore, option B is incorrect.
Option C (12): The only four-digit number whose digits add up to 12 is 1233. However, the product of its digits is 1 x 2 x 3 x 3 = 18, which is not equal to 12. Therefore, option C is incorrect.
Option D (14): The only four-digit number whose digits add up to 14 is 1124. However, the product of its digits is 1 x 1 x 2 x 4 = 8, which is not equal to 14. Therefore, option D is incorrect.
Option A (8): The only four-digit number whose digits add up to 8 is 1124. The product of its digits is 1 x 1 x 2 x 4 = 8, which is equal to the sum of the digits. This satisfies the second condition. Also, the sum of the digits (1 + 1 + 2 + 4 = 8) divides the number (1124) completely, satisfying the third condition. Therefore, the correct answer is option A.
Hence, we can conclude that the sum of the digits of the number is 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 option 'A'. Can you explain this answer?
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