Let f(x)=x^2+3x+2 and let S be the set of integers {0,1,2,…,25}. The number of members s of S such that f(s) has remainder zero when divided by 6 is:

1964 AMC 12 Problem 16

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16.

Let

f(x)=x^2+3x+2

and let

S

be the set of integers

\{0,1,2,\ldots,25\}.

The number of members

s

S

such that

f(s)

has remainder zero when divided by

6

is:

25

22

21

18

17

Factor

f(s)

(s+1)(s+2)

The product is always even; determine which residue of

s

modulo

3

fails

We have

f(s)=(s+1)(s+2),

a product of consecutive integers, so it is always even. It is divisible by

3

unless

s\equiv0\pmod3.

Among

0,1,\ldots,25,

nine values are multiples of

3.

Thus

26-9=17

values work.

Therefore, the correct answer is E.