yellowho wrote:I. The least/smallest integer is odd
II. None of the integers is divisible by 5
III. One of the integers has 3 as its units digit

If the sum of 4 consecutive integers is divisible by 5, which of the following must be true?

A) I only
B) II only
C) III only
D) I and III
E) None of the above

How are people breaking this down? Especially III.

For the sum to be a multiple of 5, the units digit of the sum must be 0 or 5. Thus, we need to determine which combinations of consecutive digits will yield a sum whose units digit is 0 or 5.

1+2+3+4 = 10. This works.
2+3+4+5 = 14. Doesn't work.
3+4+5+6 = 18. Doesn't work.
4+5+6+7 = 22. Doesn't work.
5+6+7+8 = 26. Doesn't work.
6+7+8+9 = 30. This works.
7+8+9+0 = 24. Doesn't work.
8+9+0+1 = 18. Doesn't work.
9+0+1+2 = 12. Doesn't work.
0+1+2+3 = 6. Doesn't work.

Only the following combinations work: 1, 2, 3, 4 and 6, 7, 8, 9.
Now onto the answer choices:

I. The smallest integer doesn't have to be odd. The correct answer choice cannot include I. Eliminate A and D.

II. Since any integer that has a units digit of 1, 2, 3, 4, 6, 7, 8, or 9 is not a multiple of 5, it must be true that none of the integers is divisible by 5. The correct answer choice must include II. Eliminate C and E.

The correct answer is B.

OA is actually E. I also got B. OA is wrong?

[quote="Rahul@gurome"][quote="yellowho"]If the sum of 4 consecutive integers is divisible by 5, which of the following must be true?

I. The least/smallest integer is odd
II. None of the integers is divisible by 5
III. One of the integers has 3 as its units digit[/quote]

Say the consecutive integers are n, (n + 1), (n + 2), and (n + 3)
Therefore, sum = (4n + 6)

Thus, (4n + 6) is divisible by 5.
=> (4n + 1) is divisible by 5 ------------------- (i)
=> [5n - (n - 1)] is divisible by 5
=> (n - 1) is divisible by 5 --------------------- (ii)

From (i), we see that (4n + 1) can be even or odd. Hence n can be even or odd too. Thus (I) is not always true.

From (ii), we see that (n - 1) is divisible by 5. Hence the next four integers, i.e. n, (n + 1), (n + 2), and (n + 3) are not divisible by 5. Thus (II) must is always true.

Also from (ii) we can see that last digit of (n - 1) is either 0 or 5. Accordingly the last digits of n, (n + 1), (n + 2), and (n + 3) are either (1, 2, 3, 4) or (6, 7, 8, 9). Hence (III) is not always true.

[spoiler]Thus the correct answer is B.[/spoiler][/quote]