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If the 5-digit number 750PQ is divisible by 3, 7 and 11, then what is the value of P + 2Q?
- a)17
- b)15
- c)18
- d)16
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
If the 5-digit number 750PQ is divisible by 3, 7 and 11, then what is ...
Given:
Five-digit number 750PQ is divisible by 3, 7 and 11
Concept used:
Concept of LCM
Calculation:
The LCM of 3, 7, and 11 is 231.
By taking the largest 5-digit number 75099 and dividing it by 231.
If we divide 75099 by 231 we get 325 as the quotient and 24 as the remainder.
Then, the five-digit number is 75099 - 24 = 75075.
The number = 75075 and P = 7, Q = 5
now,
P + 2Q = 7 + 10 = 17
∴ The value of P + 2Q is 17.
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Community Answer
If the 5-digit number 750PQ is divisible by 3, 7 and 11, then what is ...
To find the value of P and 2Q, we need to determine the possible values for P and Q that satisfy the given conditions.
Divisibility by 3:
A number is divisible by 3 if the sum of its digits is divisible by 3. In the given number 750PQ, the sum of the digits is 7 + 5 + 0 + P + Q = 12 + P + Q. For this number to be divisible by 3, 12 + P + Q must be divisible by 3.
Divisibility by 7:
To determine if a number is divisible by 7, we can use the divisibility rule. We subtract twice the unit digit from the remaining digits, and if the result is divisible by 7, then the number is also divisible by 7. In the given number 750PQ, we have 750P - 2Q. For this number to be divisible by 7, 750P - 2Q must be divisible by 7.
Divisibility by 11:
To determine if a number is divisible by 11, we can use the divisibility rule. We subtract the unit digit from the remaining digits, and if the result is divisible by 11, then the number is also divisible by 11. In the given number 750PQ, we have 750P - Q. For this number to be divisible by 11, 750P - Q must be divisible by 11.
Since the number is divisible by all three numbers, we can write the following equations:
12 + P + Q is divisible by 3
750P - 2Q is divisible by 7
750P - Q is divisible by 11
From the first equation, we know that 12 + P + Q is divisible by 3. The only possible values for P and Q that satisfy this condition are P = 1 and Q = 2.
Substituting these values into the second equation, we have:
750(1) - 2(2) = 748, which is not divisible by 7.
Therefore, the values P = 1 and Q = 2 do not satisfy all the given conditions.
Hence, the correct answer is option 'A' (17).
Divisibility by 3:
A number is divisible by 3 if the sum of its digits is divisible by 3. In the given number 750PQ, the sum of the digits is 7 + 5 + 0 + P + Q = 12 + P + Q. For this number to be divisible by 3, 12 + P + Q must be divisible by 3.
Divisibility by 7:
To determine if a number is divisible by 7, we can use the divisibility rule. We subtract twice the unit digit from the remaining digits, and if the result is divisible by 7, then the number is also divisible by 7. In the given number 750PQ, we have 750P - 2Q. For this number to be divisible by 7, 750P - 2Q must be divisible by 7.
Divisibility by 11:
To determine if a number is divisible by 11, we can use the divisibility rule. We subtract the unit digit from the remaining digits, and if the result is divisible by 11, then the number is also divisible by 11. In the given number 750PQ, we have 750P - Q. For this number to be divisible by 11, 750P - Q must be divisible by 11.
Since the number is divisible by all three numbers, we can write the following equations:
12 + P + Q is divisible by 3
750P - 2Q is divisible by 7
750P - Q is divisible by 11
From the first equation, we know that 12 + P + Q is divisible by 3. The only possible values for P and Q that satisfy this condition are P = 1 and Q = 2.
Substituting these values into the second equation, we have:
750(1) - 2(2) = 748, which is not divisible by 7.
Therefore, the values P = 1 and Q = 2 do not satisfy all the given conditions.
Hence, the correct answer is option 'A' (17).
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