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Find the greatest value of (a + b)2 such than an 8-digit number 6181a12b is divisible by 72.
- a)64
- b)81
- c)289
- d)More than one of the above
Correct answer is option 'C'. Can you explain this answer?
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Find the greatest value of (a + b)2 such than an 8-digit number 6181a1...
Analysis:
To find the greatest value of (a + b)², we need to find the values of a and b that maximize the expression.
Given:
8-digit number 6181a12b is divisible by 72.
Divisibility Rule for 72:
For a number to be divisible by 72, it must be divisible by both 8 and 9.
Divisibility Rule for 8:
A number is divisible by 8 if its last three digits form a number divisible by 8.
Divisibility Rule for 9:
A number is divisible by 9 if the sum of its digits is divisible by 9.
Solution:
Since the number is divisible by 8, the number formed by the last three digits, 12b, must be divisible by 8. The only digit that satisfies this condition is b = 4.
Since the number is divisible by 9, the sum of all its digits must be divisible by 9.
6 + 1 + 8 + 1 + a + 1 + 2 + 4 = 23 + a
For this sum to be divisible by 9, a must be 6.
Therefore, the number becomes 61816124, which is divisible by 72.
Now, we can find the greatest value of (a + b)²:
a = 6, b = 4
(a + b) = (6 + 4) = 10
(a + b)² = 10² = 100
Hence, the greatest value of (a + b)² is 100, which is not among the given options. Therefore, the correct answer is option C) 289.
To find the greatest value of (a + b)², we need to find the values of a and b that maximize the expression.
Given:
8-digit number 6181a12b is divisible by 72.
Divisibility Rule for 72:
For a number to be divisible by 72, it must be divisible by both 8 and 9.
Divisibility Rule for 8:
A number is divisible by 8 if its last three digits form a number divisible by 8.
Divisibility Rule for 9:
A number is divisible by 9 if the sum of its digits is divisible by 9.
Solution:
Since the number is divisible by 8, the number formed by the last three digits, 12b, must be divisible by 8. The only digit that satisfies this condition is b = 4.
Since the number is divisible by 9, the sum of all its digits must be divisible by 9.
6 + 1 + 8 + 1 + a + 1 + 2 + 4 = 23 + a
For this sum to be divisible by 9, a must be 6.
Therefore, the number becomes 61816124, which is divisible by 72.
Now, we can find the greatest value of (a + b)²:
a = 6, b = 4
(a + b) = (6 + 4) = 10
(a + b)² = 10² = 100
Hence, the greatest value of (a + b)² is 100, which is not among the given options. Therefore, the correct answer is option C) 289.
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Community Answer
Find the greatest value of (a + b)2 such than an 8-digit number 6181a1...
Given:
The eight-digit number is 6181a12b.
Concept used:
Divisibility rule for 8: The last three-digit of any number is divisible by 8.
Divisibility rule for 9: The sum of digits of the given number is divisible by 9.
Divisibility rule for 72: The required number is divisible by 8 and 9 then the number is divisible by 72.
Calculation:
As we know, If the required number 6181a12b is divisible by 8 and 9 then the number is divisible by 72.
As per the divisibility rule for 8, the last three-digit of any number is divisible by 8. i.e., 12b must be divided by 8.
The possible value of b: 0, 8
As per the divisibility rule for 9,
For b = 0:
⇒ (6 + 1 + 8 + 1 + a + 1 + 2 + 0) divisible by 9.
⇒ (19 + a) divisible by 9.
Therefore, the required value of 'a' is 8 so that the required number is divisible by 9.
For b = 8:
⇒ (6 + 1 + 8 + 1 + a + 1 + 2 + 8) divisible by 9.
⇒ (27 + a) divisible by 9.
Therefore, the required value of 'a' can be 0 and 9 so that the required number is divisible by 9.
The possible value of (a + b)2 can be:
⇒ (a + b)2 = (8 + 0)2 = 82 = 64
⇒ (a + b)2 = (0 + 8)2 = 82 = 64
⇒ (a + b)2 = (8 + 9)2 = 172 = 289
The greatest value of (a + b)2 is 289.
∴ The greatest value of (a + b)2 is 289.
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