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If x is a positive integer, is x divided by 5 an odd integer?
 x contains only odd factors
 x is a multiple of 5
Correct Answer: C
A number divided by 5 will be an odd integer if and only if that number contains only odd factors, one of which is 5.
 A number divided by 5 will be an odd integer if and only if that number contains only odd factors, one of which is 5. In other words, there are two conditions under which x divided by 5 will be an odd integer:
(1) x is a multiple of 5
(2) x contains only odd factors 
Evaluate Statement (1) alone.
 If x contains only odd factors, there is no guarantee that one of those factors is 5. Consequently, there is no guarantee that x will be divisible by 5.
 For example:
9 = 3*3 > not an odd integer when divided by 5 since 5 is not a factor
21 = 3*7 > not an odd integer when divided by 5 since 5 is not a factor
15 = 3*5 > an odd integer when divided by 5 since 5 is a factor
105 = 3*7*5 > an odd integer when divided by 5 since 5 is a factor  Statement (1) alone is NOT SUFFICIENT.

Evaluate Statement (2) alone.
 Simply because x is a multiple of 5 does not guarantee that x only contains odd factors. Consequently, there is no guarantee that x is divisible by 5.
5 = 5*1 > an odd integer when divided by 5 because 5 is a factor and there are only odd factors
15 = 5*3 > an odd integer when divided by 5 because 5 is a factor and there are only odd factors
25 = 5*5 > an odd integer when divided by 5 because 5 is a factor and there are only odd factors
20 = 5*4 > not an odd integer when divided by 5 because there is at least one even factor
30 = 6*5 > not an odd integer when divided by 5 because there is at least one even factor  Statement (2) alone is NOT SUFFICIENT.
 Simply because x is a multiple of 5 does not guarantee that x only contains odd factors. Consequently, there is no guarantee that x is divisible by 5.

Evaluate Statements (1) and (2) together.
 With only odd factors and x as a multiple of 5 (i.e., with 5 as a factor), you know that x divided by 5 must be an odd number since the two conditions laid out earlier are fulfilled.
 Consider the following examples:
5 = 5*1 > an odd integer when divided by 5
15 = 5*3 > an odd integer when divided by 5
25 = 5*5 > an odd integer when divided by 5
35 = 5*7 > an odd integer when divided by 5  Statements (1) and (2), when taken together, are SUFFICIENT.
 Since Statement (1) alone is NOT SUFFICIENT and Statement (2) alone is NOT SUFFICIENT, but Statements (1) and (2), when taken together, are SUFFICIENT, answer C is correct.