Helpppppppp

Let the common odd factor be k. Since k divides x + 7, it must also divide any multiple of it, such as (x + 7)(x - 7) = x^2 - 49.

If k is a factor of both x^2 + 16 and x^2 - 49, it is mathematically required to also divide their difference:

(x^2 + 16) - (x^2 - 49) = 65

The prime factorization of 65 is 5 times 13. The odd factors (excluding 1) are 5, 13, and 65. Therefore, the original expression x + 7 must be a multiple of 5 or 13.

For natural numbers x < 100 (values from 1 to 99), we just count the possibilities:

Multiples of 5: x must end in 3 or 8 (3, 8, 13,...., 98). This gives 20 values.

• Multiples of 13: The valid values are 6, 19, 32, 45, 58, 71, 84, 97. This gives 8 values.
• Overlap: We subtract any double-counted numbers where x + 7 is a multiple of both (which is 65). This only happens when x = 58, giving us 1 overlapping value.

Total unique values = 20 + 8 - 1 = 27.