GPREP #25

[hardikrs]

For any positive integer n, the length of n is defined as the number of prime factors whose product is n. For example, the length of 75 is 3, since 75 = 3 x 5 x 5. How many two-digit positive integers have length 6?

1. None
2. One
3. Two
4. Three
5. Four

Please explain how to pick numbers or a easy way to solve this.
thanks,

[3Gs]

smallest prime no is 2.....so 2^6 = 64 ...2 digit
Next smaller no with product of 6 primes would be 2^5 * 3 = 32* 3 = 96...2 digit
Next smaller would be 2^4 * 3^2 = 16 * 9 = 144...3 digit

so only 2 such nos and answer is 3

[vineejay]

Let us consider the least prime factor 2 - and take 2^6 = 64 - possible
so 1 is not possible
now let us increase the prime factor of length 6= 2^5 * 3= 96
so 2 is not possible
now take the next = 2^4 * 3^2 = 144( not 2 digit)
so the correct answer = 3