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I guess you are asking for given the factors of 50, 51, ..., 60, which ones are odd numbers:
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 25, 27, 29, 51, 53, 55, 57, 59
Factors of the individual numbers (odd factors emboldened):
50 := 1, 2, 5, 10, 25, 50
51 := 1, 3, 17, 51
52 := 1, 2, 4, 13, 26, 52
53 := 1, 53
54 := 1, 2, 3, 6, 9, 18, 27, 54
55 := 1, 5, 11, 55
56 := 1, 2, 4, 7, 8, 14, 28, 56
57 := 1, 3, 19, 57
58 := 1, 2, 29, 58
59 := 1, 59
60 := 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Yes. Square numbers have an odd number of factors. 49 has three factors.
Even numbers must have 2 amongst their factors; they may have other even numbers amongst their factors. Odd numbers must not; every factor of an odd number must be odd.
Even powers of prime numbers. Square numbers have an odd number of factors.
Square numbers have an odd number of factors.
All odd numbers have odd factors.