How many integers less than 100 have exactly 4 odd factors but no even factors?
How many integers less than 100 have exactly 4 odd factors but no even factors?
Answer/Solution
17
Steps/Work
Two cases are possible:
x1y1=xy=odd<100x1y1=xy=odd<100, where xx and yy are odd prime numbers, then # of factors will be (1+1)(1+1)=4(1+1)(1+1)=4;
If x=3, then y can be: 5, 7, 11, 13, 17, 19, 23, 29, 31 - 9 numbers.
If x=5, then y can be: 7, 11, 13, 17, 19 - 5 numbers.
If x=7, then y can be: 11, 13 - 2 numbers.
OR:
x3=odd<100x3=odd<100, where xx is odd prime number, then # of factors will be (3+1)=4(3+1)=4.
x can be only 3 - 1 number.
Total 9+5+2+1=179+5+2+1=17 .
Answer: D.
x1y1=xy=odd<100x1y1=xy=odd<100, where xx and yy are odd prime numbers, then # of factors will be (1+1)(1+1)=4(1+1)(1+1)=4;
If x=3, then y can be: 5, 7, 11, 13, 17, 19, 23, 29, 31 - 9 numbers.
If x=5, then y can be: 7, 11, 13, 17, 19 - 5 numbers.
If x=7, then y can be: 11, 13 - 2 numbers.
OR:
x3=odd<100x3=odd<100, where xx is odd prime number, then # of factors will be (3+1)=4(3+1)=4.
x can be only 3 - 1 number.
Total 9+5+2+1=179+5+2+1=17 .
Answer: D.