consecutive positive integers?

Question

ok... i am studing for the sat. one of the questions in the sat studyguide is:

which of the following could be the remainders when four consecutive positive integers are each divided by 3?

a) 1,2,3,1
b) 1,2,3,4,
c) 0,1,2,3
d) 0,1,2,0
e) 0,2,3,0

the answer is d. but i don't understand how they got the answer. it explains it but it doesn't make sense to me. please help me.

thank you so very much!

Answer 1

I'm going to explain by using an example string of 6 integers: 1, 2, 3, 4, 5, 6.

1 / 3 = 0 r 1
2 / 3 = 0 r 2
3 / 3 = 1 r 0
4 / 3 = 1 r 1
5 / 3 = 1 r 2
6 / 3 = 2 r 0

Notice how the highest remainder is 2 and the lowest is 0. They show up in the order as in answer choice d as well.

Answer 2

All the other answers have the number "3" as part of the answer. If you divide a number by 3, you can't have a remainder of 3, because it would divide in one more time.
Example: 15/3 = 4 remainder 3, but that means it =5 instead because it goes in one more time. So pick the answer that only has 0, 1, or 2 as remainder.

Answer 3

When dividing by 3, a remainder of 3 is not possible.

A hypothetical remainder of 3 means you need to add 1 to the
quotient. By doing that you will have 0 remainder, since the
additional 3 can be divided by 3 exactly once.

This eliminates all the answers except d.

Another way of looking at this is that all multiples of 3 have a
remainder of 0. If a number is not a multiple of 3 and you divide it by 3 the remainder will be either 1 or 2.

So if you have 4 consecutive integers, the sequence of remainders will be either 1,2,0,1 or 2,0,1,2 or 0,1,2,0.
Only d fits one of these patterns, so again the answer is d.

Answer 4

At least one of the numbers is divisible by three, so at least one remainder must be 0 [rules out (a) and (b)]. No number is divisible by 3 and leaves a remained higher than 2 [rules out (c) and (e)]. The answer has to be 0-1-2-0, 1-2-0-1 or 2-0-1-2

Answer 5

The remainder of division by 3 must be less than 3. The answer must be (d) because all the others contain a remainder of 3.

Answer 6

x , x+1 , x+2 , x+3 ;
if x divided by 3 and remainder is 0 then answer is d.
if x divided by 3 and remainder is 1 then answer is d
In any case answer is d.