manpreetsingh86 wrote:
Call a number Guardian if it is composite but not divisible by 2, 3, or 5. The three smallest guardian numbers are 49, 77, and 91. There are 168 prime numbers less than 1000. How many Guardian numbers are there less than 1000?
a) 100
b) 102
c) 104
d) 106
e) 108
Dear
manpreetsingh86,
I'm happy to respond.
Here is my calculation:
Start with the numbers from 1 - 1000. We have 1000 numbers.
Subtract the primes:
1000 - 168 = 832
Now, subtract the even numbers, divisible by 2. There are 500 even numbers in the set, but the first, 2 itself, has already been subtracted with the primes. There are 499 non-prime even numbers in the set, including 1000 itself. (1000 itself really shouldn't be included, because are looking for numbers less than 1000, but it gets subtracted in this step anyway, so it doesn't matter.)
832 - 499 = 333
Now, numbers divisible by 3. There are 333 factors of three between 1 and 1000, but first of all, one of those is the prime number 3, already subtracted, so exclude that --- 332. Of those, half are even, already counted, so that leaves 332/2 = 166 non-prime odd multiples of 3.
333 - 166 = 167
Now, multiples of 5. There are 200 multiples of 5 up to 1000. Half are even, divisible by 10, so eliminate them --- 100 odd multiples of 5. Eliminate the prime number 5 itself, already counted --- 99 non-prime multiples of 5. Now, 1/3 of these are divisible by three, already counted, so we 2/3 of this number ==> 66. There are 66 non-prime multiples of 5 that are not divisible by 3.
167 - 66 = 101
The last number that needs to be subtracted --- not prime, not composite, not a multiple of 2 or 3 or 5, is 1 itself. This is not composite, so it is not a guardian.
101 - 1 = 100
That leaves 100 guardian numbers. OA =
(A) This is mathematically a very intriguing question, but I suspect that it's a notch too intricate and complicated to be something the GMAT would ask.
Mike
_________________
Mike McGarry
Magoosh Test PrepEducation is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)