Sneaky or indirect ways the GMAT uses to tell you a variable is odd

I'll go first with the ones I've seen, add yours below.

Remember, if n is odd then n is also NOT EVEN and therefore not equal to zero, 2, 4 or any even number, and does not have units digit of 0, 2, 4, 6 or 8, etc. If we know n is odd, then we can definitively answer such question as "is n even?" or "is n a multiple of 4?" or "is n a factor of 2^x (x>0)?" with NO and therefore such statements may be sufficient for DS purposes.

Knowing if n is odd also helps us answer data sufficiency questions when we have previously calculated two different values for n, one odd one even, and the question is asking us for one discreet value of n.

Some statements are sufficient on their own, some need additional information

These are all assuming that we already know that n is an integer

- n is a prime greater than 2
- n is a prime number and n is not equal to 2
- n is not a multiple of 2
- n does not have 2 as a factor
- n/2 is not an integer (must know than n is an integer!)
- n is the square of a prime number and is greater than 4
- n is the product of 2 primes that are both greater or not equal to 2 (or n is 2/4/6/any even number more/less than this product (which must be odd) May be written as something like "n+4 the product of...")
- n is one less/one more than the sum of two prime numbers greater than 2 (may also be expressed as "n+1 is...." or "n-1 is....". Anyway, the sum of two primes that are both greater than 2 is always even. If n is 1/3/5/any odd number more or less than this sum (which must be even) then n is odd)
- (-1)^n is less than zero
- (-m)^n is less than zero (we don't need to know about m. M must be positive for any value of (-m)^n to be less than zero. If m is negative, -m is positive and (-m)^n can never be less than 0)
- n is a prime number between x and y (when x is more than 2, y could be anything)
- n is a product of two prime numbers between x and y (they could give any values here but x must be more than 2)
- n^2 is an odd number
- n^3 is an odd number
- n^x is an odd number and x is not equal to zero (Careful!! n^x will always be odd if n is odd. However, if x is 0, n^x will be equal to 1, and is therefore odd when n is both even or odd. This is a sneaky way a DS question can trick you into thinking n is odd, when it might not be. Unless you know that x is not equal to zero, "n^x is odd" does not necessarily mean n is odd and alone is insufficient to determine whether n is odd.)
- the units digit of n is a prime number greater than 2
- 2^n has a units digit of 2 (this means that n is 1, 5, 9 etc - always odd)
- 2^n has a units digit of 8 (this means that n is 3, 7, 11 etc - always odd)
- 9^n has a units digit of 9 (same theme. There are several variations on this)
- m*n is odd (we must know that m and n are integers)
- m^n = m, and m is not equal to 0 or 1 (If m is -1, n can be any odd number. For other values of m, n must be equal to 1, which is odd. The question may express the conditions as some thing like m>1)
- n/3 is an odd integer (this tells you that n is an odd multiple of 3) (there can be many variations on this. To generalise, if the statement gives you any variation of "n^x/y = an odd integer" and y is an odd integer and x is not zero, we know n must be odd)
- m+n is an even number and m is odd (might be told that m is odd in any of the ways above, or given a direct value of m e.g. m=5, or given information that allows the calculation of m)

And some phrases used that on their own do NOT definitively tell you whether n is odd or even
- 1/n is a terminating decimal (1/5 and 1/2 are both terminating)
- 1/n is a repeating decimal (1/3 and 1/6 are both repeating)
- the reciprocal of the units digit of n is a repeating decimal/terminating decimal (see above. The question may then go on to tell you something else, such as "5 is not a factor of n", that allows you to determine the u.d. of n)
- n^x is an odd number (see above. X could be equal to zero, in which case n^x =1 for all values of n, odd and even)
- (-m)^n is positive (unlike the example above, in this case we do need to know if m is positive or negative, and cannot assume)
- n is prime
- 6^n has a units digit of 6/5^n has a units digit of 5
- 1^n is odd (1^n = 1 for all values of n, so it's always odd)
- m^n = m (this phrase tricks you into thinking that maybe n=1, but in fact we could have m=1 and any value of n, including zero)
- n is the product of two prime numbers
- n is the sum of two prime numbers