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Suppose A and B represent natural numbers less than 10 and the number '9874A67B' is divisible by both 9 and 11. What is the product of A and B?
- a)36
- b)63
- c)40
- d)Can't be determined
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
Suppose A and B represent natural numbers less than 10 and the number...
Divisibility rule of 9: 9+8+7+4+6+7+A+B is divisible by 9 (so A+B should equal 4 or 13)
Divisibility rule of 11: |9+7+A+7| - |8+4+6+B| = 11 ⇒ A+5-B or B-A+5 is a multiple of 11..
The two divisibility rules are solvable only if A = 4 and B = 9.
The product of A and B is 36.
So, answer is Option A.
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Community Answer
Suppose A and B represent natural numbers less than 10 and the number...
To find the product of A and B, we need to first determine the possible values of A and B.
For a number to be divisible by 9, the sum of its digits must be divisible by 9. In the given number '9874A67B', we know that the sum of the digits is divisible by 9. Since the sum of the digits from 1 to 9 is 45, the sum of the digits A and B must be 9 in order for the number to be divisible by 9.
For a number to be divisible by 11, the difference between the sum of the digits at even positions and the sum of the digits at odd positions must be divisible by 11. In the given number '9874A67B', the sum of the digits at even positions is 9 + 7 + 7 = 23, and the sum of the digits at odd positions is 8 + 4 + A + B = 12 + A + B. Therefore, the difference between these two sums is (12 + A + B) - 23 = A + B - 11.
Since the number '9874A67B' is divisible by 11, the difference between the sums of the digits at even and odd positions must be divisible by 11. Therefore, A + B - 11 must be divisible by 11.
Combining the two conditions, we have A + B = 9 and A + B - 11 = 0 (mod 11).
Solving these equations simultaneously, we get A = 5 and B = 4.
Therefore, the product of A and B is 5 * 4 = 20, which is not one of the given options.
Since none of the given options match the correct answer, the correct answer is "Can't be determined" (option D).
For a number to be divisible by 9, the sum of its digits must be divisible by 9. In the given number '9874A67B', we know that the sum of the digits is divisible by 9. Since the sum of the digits from 1 to 9 is 45, the sum of the digits A and B must be 9 in order for the number to be divisible by 9.
For a number to be divisible by 11, the difference between the sum of the digits at even positions and the sum of the digits at odd positions must be divisible by 11. In the given number '9874A67B', the sum of the digits at even positions is 9 + 7 + 7 = 23, and the sum of the digits at odd positions is 8 + 4 + A + B = 12 + A + B. Therefore, the difference between these two sums is (12 + A + B) - 23 = A + B - 11.
Since the number '9874A67B' is divisible by 11, the difference between the sums of the digits at even and odd positions must be divisible by 11. Therefore, A + B - 11 must be divisible by 11.
Combining the two conditions, we have A + B = 9 and A + B - 11 = 0 (mod 11).
Solving these equations simultaneously, we get A = 5 and B = 4.
Therefore, the product of A and B is 5 * 4 = 20, which is not one of the given options.
Since none of the given options match the correct answer, the correct answer is "Can't be determined" (option D).
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Suppose A and B represent natural numbers less than 10 and the number '9874A67B' is divisible by both 9 and 11. What is the product of A and B?a)36b)63c)40d)Can't be determinedCorrect answer is option 'A'. Can you explain this answer?
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