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If the nine-digit number 7p5964q28 is completely divisible by 88, what is the value of (p2 − q), for the largest value of q, where p and q are natural numbers?
Correct answer is '72'. Can you explain this answer?
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Verified Answer
If the nine-digit number 7p5964q28 is completely divisible by 88, wha...
Number should be divisible by 88 then number should be divisible by 8 and 11 both. Largest possible value of q to be divisible by 98 is = 9
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⇒ 92/88 = 116
For q = 9, then
Number should be divisible by 11
= (7 + 5 + 6 + 9 + 8) − (p + 9 + 4 + 2)
= 20 − p
(20 − p) should be divisible by 11 then for p = 9 this condition will satisfy.
So, possible value of p = 9
Then,
∴ p2 − q = 81 − 9 = 72
Hence, the correct answer is 72.
Most Upvoted Answer
If the nine-digit number 7p5964q28 is completely divisible by 88, wha...
To determine the value of (p2 − q), we need to find the largest possible value of q for which the nine-digit number 7p5964q28 is divisible by 88.
To solve this problem, we need to understand the divisibility rules for 88 and how to manipulate the given number to meet those rules.
1. Divisibility Rule for 88:
A number is divisible by 88 if the number formed by its last three digits is divisible by 8 and the number formed by its last two digits is divisible by 11.
2. Divisibility Rule for 8:
A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
3. Divisibility Rule for 11:
A number is divisible by 11 if the difference between the sums of its alternate digits is divisible by 11.
Let's apply these rules step by step to find the largest possible value of q.
Step 1: Divisibility by 8
From the given number, the last three digits are 628. To check if it is divisible by 8, we need to see if 628 is divisible by 8. Since 628 is not divisible by 8, we need to increase q.
Step 2: Divisibility by 11
Next, we need to check if the difference between the sums of the alternate digits is divisible by 11. The alternate digits are 7, 5, 9, 4, and 2. The sum of the digits in the odd positions is 7 + 9 + 2 = 18, and the sum of the digits in the even positions is 5 + 4 = 9. The difference between these sums is 18 - 9 = 9, which is not divisible by 11. Therefore, we need to increase q further.
Step 3: Increasing q
To make the number divisible by 8, we need to increase q to the next multiple of 8. The nearest multiple of 8 greater than 628 is 632. So, we update q = 32.
Step 4: Increasing q to meet the divisibility by 11
To make the number divisible by 11, we need to increase q to the next multiple of 11. The nearest multiple of 11 greater than 32 is 44. However, we need to find the largest value of q, so we backtrack and try the next multiple of 11, which is 33. By increasing q to 33, the sum of the alternate digits becomes 7 + 9 + 3 = 19, and the sum of the even-positioned digits remains 9. The difference between these sums is 19 - 9 = 10, which is divisible by 11.
Therefore, the largest value of q is 33, and the value of (p2 − q) = (p2 − 33).
To solve this problem, we need to understand the divisibility rules for 88 and how to manipulate the given number to meet those rules.
1. Divisibility Rule for 88:
A number is divisible by 88 if the number formed by its last three digits is divisible by 8 and the number formed by its last two digits is divisible by 11.
2. Divisibility Rule for 8:
A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
3. Divisibility Rule for 11:
A number is divisible by 11 if the difference between the sums of its alternate digits is divisible by 11.
Let's apply these rules step by step to find the largest possible value of q.
Step 1: Divisibility by 8
From the given number, the last three digits are 628. To check if it is divisible by 8, we need to see if 628 is divisible by 8. Since 628 is not divisible by 8, we need to increase q.
Step 2: Divisibility by 11
Next, we need to check if the difference between the sums of the alternate digits is divisible by 11. The alternate digits are 7, 5, 9, 4, and 2. The sum of the digits in the odd positions is 7 + 9 + 2 = 18, and the sum of the digits in the even positions is 5 + 4 = 9. The difference between these sums is 18 - 9 = 9, which is not divisible by 11. Therefore, we need to increase q further.
Step 3: Increasing q
To make the number divisible by 8, we need to increase q to the next multiple of 8. The nearest multiple of 8 greater than 628 is 632. So, we update q = 32.
Step 4: Increasing q to meet the divisibility by 11
To make the number divisible by 11, we need to increase q to the next multiple of 11. The nearest multiple of 11 greater than 32 is 44. However, we need to find the largest value of q, so we backtrack and try the next multiple of 11, which is 33. By increasing q to 33, the sum of the alternate digits becomes 7 + 9 + 3 = 19, and the sum of the even-positioned digits remains 9. The difference between these sums is 19 - 9 = 10, which is divisible by 11.
Therefore, the largest value of q is 33, and the value of (p2 − q) = (p2 − 33).
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If the nine-digit number 7p5964q28 is completely divisible by 88, what is the value of (p2 − q), for the largest value of q, where p and q are natural numbers?Correct answer is '72'. Can you explain this answer?
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If the nine-digit number 7p5964q28 is completely divisible by 88, what is the value of (p2 − q), for the largest value of q, where p and q are natural numbers?Correct answer is '72'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about If the nine-digit number 7p5964q28 is completely divisible by 88, what is the value of (p2 − q), for the largest value of q, where p and q are natural numbers?Correct answer is '72'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the nine-digit number 7p5964q28 is completely divisible by 88, what is the value of (p2 − q), for the largest value of q, where p and q are natural numbers?Correct answer is '72'. Can you explain this answer?.
If the nine-digit number 7p5964q28 is completely divisible by 88, what is the value of (p2 − q), for the largest value of q, where p and q are natural numbers?Correct answer is '72'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about If the nine-digit number 7p5964q28 is completely divisible by 88, what is the value of (p2 − q), for the largest value of q, where p and q are natural numbers?Correct answer is '72'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the nine-digit number 7p5964q28 is completely divisible by 88, what is the value of (p2 − q), for the largest value of q, where p and q are natural numbers?Correct answer is '72'. Can you explain this answer?.
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