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How many numbers are there less than 100 that cannot be written as a multiple of a perfect square greater than 1 ?
- a)61
- b)56
- c)52
- d)65
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
How many numbers are there less than 100 that cannot be written as a m...
To begin with, all prime numbers will be part of this list.
There are 25 primes less than 100 .
Apart from this, any number that can be written as a product of two or more primes will be there on this list.
That is, any number of the form pq, or pqr, or pqrs will be there on this list (where p,q,r,s are primes).
A number of the form pnq cannot be a part of this list if n is greater than 1, as then the number will be a multiple of p2
First let us think of all multiples of 2× prime number. This includes 2×3,2×5,2×7,2×11 all the way up to 2×47(14 numbers).
The, we move on to all numbers of the type 3∗ prime number 3×5,3×7 all the way up to 3×31(9 numbers).
Then, all numbers of the type 5× prime number −5×7,5×11,5×13,5×17,5×19(5 numbers).
Then, all numbers of the type 7× prime number and then 7×11,7×13 (2 numbers).
There are no numbers of the form 11× prime number which have not been counted earlier.
Post this, we need to count all numbers of the form p×q×r, where p,q,r are all prime.
Adding 1 to this list, we get totally 36 different composite numbers.
Along with the 25 prime numbers, we get 61 numbers that cannot be written as a product of a perfect square greater than 1 .
Hence the answer is 61
There are 25 primes less than 100 .
Apart from this, any number that can be written as a product of two or more primes will be there on this list.
That is, any number of the form pq, or pqr, or pqrs will be there on this list (where p,q,r,s are primes).
A number of the form pnq cannot be a part of this list if n is greater than 1, as then the number will be a multiple of p2
First let us think of all multiples of 2× prime number. This includes 2×3,2×5,2×7,2×11 all the way up to 2×47(14 numbers).
The, we move on to all numbers of the type 3∗ prime number 3×5,3×7 all the way up to 3×31(9 numbers).
Then, all numbers of the type 5× prime number −5×7,5×11,5×13,5×17,5×19(5 numbers).
Then, all numbers of the type 7× prime number and then 7×11,7×13 (2 numbers).
There are no numbers of the form 11× prime number which have not been counted earlier.
Post this, we need to count all numbers of the form p×q×r, where p,q,r are all prime.
Adding 1 to this list, we get totally 36 different composite numbers.
Along with the 25 prime numbers, we get 61 numbers that cannot be written as a product of a perfect square greater than 1 .
Hence the answer is 61
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Community Answer
How many numbers are there less than 100 that cannot be written as a m...
Counting the numbers less than 100:
To find the numbers that cannot be written as a multiple of a perfect square greater than 1, we need to count the numbers less than 100 that do not have any perfect square greater than 1 as a factor.
Finding the perfect squares:
First, we need to identify the perfect squares less than 100. The perfect squares less than 100 are: 1, 4, 9, 16, 25, 36, 49, 64, 81.
Counting the numbers:
To count the numbers that cannot be written as a multiple of a perfect square greater than 1, we need to subtract the numbers that are divisible by a perfect square greater than 1 from the total count of numbers less than 100.
There are 99 numbers less than 100.
Out of these, we need to subtract the numbers that are divisible by 4, 9, 16, 25, 36, 49, 64, and 81.
Calculating the count:
To calculate the count, we can use the principle of inclusion-exclusion.
Numbers divisible by 4:
The numbers divisible by 4 are: 4, 8, 12, ..., 96. There are 24 numbers divisible by 4.
Numbers divisible by 9:
The numbers divisible by 9 are: 9, 18, 27, ..., 90. There are 10 numbers divisible by 9.
Numbers divisible by 16:
The numbers divisible by 16 are: 16, 32, 48, 64, 80. There are 5 numbers divisible by 16.
Numbers divisible by 25:
The numbers divisible by 25 are: 25, 50, 75. There are 3 numbers divisible by 25.
Numbers divisible by 36:
The numbers divisible by 36 are: 36, 72. There are 2 numbers divisible by 36.
Numbers divisible by 49:
The numbers divisible by 49 are: 49. There is 1 number divisible by 49.
Numbers divisible by 64:
The numbers divisible by 64 are: 64. There is 1 number divisible by 64.
Numbers divisible by 81:
The numbers divisible by 81 are: 81. There is 1 number divisible by 81.
Calculating the count:
Now, we can calculate the count by subtracting the numbers divisible by each perfect square from the total count of numbers.
Count = Total count - Numbers divisible by 4 - Numbers divisible by 9 - Numbers divisible by 16 - Numbers divisible by 25 - Numbers divisible by 36 - Numbers divisible by 49 - Numbers divisible by 64 - Numbers divisible by 81
Count = 99 - 24 - 10 - 5 - 3 - 2 - 1 - 1 - 1
Count = 99 - 48
Count = 51
Therefore, there are 51 numbers less than 100 that cannot be written as a multiple of a perfect square greater than 1. The correct answer is option 'A' (
To find the numbers that cannot be written as a multiple of a perfect square greater than 1, we need to count the numbers less than 100 that do not have any perfect square greater than 1 as a factor.
Finding the perfect squares:
First, we need to identify the perfect squares less than 100. The perfect squares less than 100 are: 1, 4, 9, 16, 25, 36, 49, 64, 81.
Counting the numbers:
To count the numbers that cannot be written as a multiple of a perfect square greater than 1, we need to subtract the numbers that are divisible by a perfect square greater than 1 from the total count of numbers less than 100.
There are 99 numbers less than 100.
Out of these, we need to subtract the numbers that are divisible by 4, 9, 16, 25, 36, 49, 64, and 81.
Calculating the count:
To calculate the count, we can use the principle of inclusion-exclusion.
Numbers divisible by 4:
The numbers divisible by 4 are: 4, 8, 12, ..., 96. There are 24 numbers divisible by 4.
Numbers divisible by 9:
The numbers divisible by 9 are: 9, 18, 27, ..., 90. There are 10 numbers divisible by 9.
Numbers divisible by 16:
The numbers divisible by 16 are: 16, 32, 48, 64, 80. There are 5 numbers divisible by 16.
Numbers divisible by 25:
The numbers divisible by 25 are: 25, 50, 75. There are 3 numbers divisible by 25.
Numbers divisible by 36:
The numbers divisible by 36 are: 36, 72. There are 2 numbers divisible by 36.
Numbers divisible by 49:
The numbers divisible by 49 are: 49. There is 1 number divisible by 49.
Numbers divisible by 64:
The numbers divisible by 64 are: 64. There is 1 number divisible by 64.
Numbers divisible by 81:
The numbers divisible by 81 are: 81. There is 1 number divisible by 81.
Calculating the count:
Now, we can calculate the count by subtracting the numbers divisible by each perfect square from the total count of numbers.
Count = Total count - Numbers divisible by 4 - Numbers divisible by 9 - Numbers divisible by 16 - Numbers divisible by 25 - Numbers divisible by 36 - Numbers divisible by 49 - Numbers divisible by 64 - Numbers divisible by 81
Count = 99 - 24 - 10 - 5 - 3 - 2 - 1 - 1 - 1
Count = 99 - 48
Count = 51
Therefore, there are 51 numbers less than 100 that cannot be written as a multiple of a perfect square greater than 1. The correct answer is option 'A' (
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