Consider the odd numbers 4, 9, 16, 36, 49, 64, and 81 are called square numbers.

1. Consider the odd square numbers less than 100. How do their factor sums differ from the factor sums of non-square odd numbers?

2. Consider the even square numbers less than 100. How do their factor sums differ from the factor sums of non-square even numbers?

1) The sum of the odd square numbers less than 100 are all odd while the factor sums of non-square odd numbers are also always odd.

2) The sum of the even square numbers less than 100 are always odd while the sum of of non-square even numbers are not always odd.

Step-by-step explanation:

The odd square numbers are:

9,49,81

The factors are:

9={1,3,9}

Sum of factors of 9:1+3+9=13

The factors of 49 are:

49={1,7,49}

Sum of factors of 49 : 1+7+49=57

The factors of 81 are: {1,3,9,27,81}

Their sum:1+3+9+27+81=121

The non-square even numbers are:

2,6,8,10,12,14, ...

Factors of 2:{1,2}

Sum of factors:1+2=3

Factors of 6: {1,2,3,6}

Sum of factors:1+2+3+6=12

Factors of 8: {1,2,4,8}

Sum: 1+2+4+8=15

Factors of 10:{1,2,5,10}

Sum: 1+2+5+10=18

Factors of 12:{1,2,3,4,6,12}

Sum:1+2+3+4+6+12=28

Factors of 14: {1,2,7,14}

Sum:1+2+7+14=24

Even square numbers less than 100 are:

4,16,36,64

Factors of 4={1,2,4}

Sum:1+2+4=7

Factors of 16: {1,2,4,8,16}

Sum : 1+2+4+8+16=31

Factors of 36: {1,2,3,4,6,9,12,18,36}

Sum : 1+2+3+4+6+9+12+18+36=91

The non-square odd numbers are:

3,5,7,11, ...

Factors of 3:{1,3}

Sum : 1+3=4

Factors of 5: {1,5}

Sum: 1+5=6

Factors of 7:{1,7}

Sum of factors: 1+7=8

Factors of 15: {1,3,5,15}

Sum: 1+3+5+15=24

Factors of 21:{1,3,7,21}

Sum: 1+3+7+21=32

Factors of 27: {1,3,9,27}

Sum: 1+3+9+27=40