2. If \( c \) is any positive integer, which of the following expressions must be an odd integer F. \( \frac{c}{7} \) G. \( c+7 \) H. \( 7 c \) J. \( c^{7} \) (K. \( 7^{c} \) \( 7^{2}=49 \)

To determine which of the given expressions must be an odd integer when \( c \) is any positive integer, let's analyze each option step by step. ### Option F: \( \frac{c}{7} \) - This expression will only be an integer if \( c \) is a multiple of 7. If \( c \) is not a multiple of 7, \( \frac{c}{7} \) will not be an integer. Therefore, it cannot be guaranteed to be an odd integer for all positive integers \( c \). ### Option G: \( c + 7 \) - The sum of an odd integer and an even integer is odd, and the sum of two odd integers is even. Since \( c \) can be either odd or even, \( c + 7 \) can be either odd or even depending on the parity of \( c \). Thus, this expression does not have to be an odd integer. ### Option H: \( 7c \) - The product of an odd integer (7) and any integer \( c \) will always be odd if \( c \) is odd, and even if \( c \) is even. Therefore, \( 7c \) can be either odd or even depending on \( c \). This expression does not have to be an odd integer. ### Option I: \( c^{7} \) - The power of an odd integer is always odd, and the power of an even integer is always even. Since \( c \) can be either odd or even, \( c^{7} \) can also be either odd or even. Thus, this expression does not have to be an odd integer. ### Option J: \( 7^{c} \) - The base 7 is an odd integer. Any power of an odd integer is also odd. Therefore, \( 7^{c} \) will always be an odd integer for any positive integer \( c \). ### Conclusion The only expression that must be an odd integer for any positive integer \( c \) is: **J. \( 7^{c} \)**