The number of positive integers which can be formed by using any number of digits from 0, 1, 2, 3, 4, 5 but using each digit not more than once in each number, is

1) 1200

2) 1500

3) 1600

4) 1630  

A pretti number is a seven digit positive integer with following properties the integer formed by its leftmost three digits is a perfect square the integer formed by its rightmost 4 digits is a perfect cube its ten thousands digit and ones (units) digits are equal its thousands digit is not zero

How many pretti numbers are there?

 A Pretti number is a seven-digit positive integer with the following properties:
• The integer formed by its leftmost three digits is a perfect square.
• The integer formed by its rightmost four digits is a perfect cube.
• Its ten thousands digit and ones (units) digit are equal.
• Its thousands digit is not zero.
How many Pretti numbers are there?

If an integer n is written as a product of prime numbers, this product (known as its
prime factorization) can be used to determine the number of positive factors of n. For
example, the prime factorization of 28 = 2 × 2 × 7 = 22 × 7
1
. The positive factors of 28
are:
28 = 22 × 7
1
14 = 21 × 7
1
7 = 20 × 7
1
4 = 22 × 7
0
2 = 21 × 7
0
1 = 20 × 7
0
Each positive factor includes 2, 1 or 0 twos, 1 or 0 sevens, and no other prime numbers.
Since there are 3 choices for the number of twos, and 2 choices for the number of sevens,
there are 3 × 2 = 6 positive factors of 28.
(a) How many positive factors does 675 have?
(b) A positive integer n has the positive factors 9, 11, 15, and 25 and exactly fourteen
other positive factors. Determine the value of n.
(c) Determine the number of positive integers less than 500 that have the positive
factors 2 and 9 and exactly ten other positive factors.