The cookies in a jar contain a total of 1000 chocolate chips. All but one of these cookies contains the same number of chips; it contains one more chip than the others. The number of cookies in the jar is between one dozen and three dozen. What is the sum of the number of cookies in the jar and the number of chips in the cookie with the extra chocolate chip?
(A) 65
(B) 64
(C) 63
(D) 66
(E) 67
The cookies in a jar contain a total of 1000 chocolate chips. All but one of these cookies contains the same number of chips; it contains one more chip than the others. The number of cookies in the jar is between one dozen and three dozen. What is the sum of the number of cookies in the jar and the number of chips in the cookie with the extra chocolate chip?
(A) 65
(B) 64
(C) 63
(D) 66
(E) 67

## Solution
25.
If we remove the extra chip from the special cookie, all cookies have the same number of chocolate chips for a total of 999 chips. We look at factorizations of 999 .

The question states that the number of cookies in the jar is between 12 and 36 so this implies that the only factorization of 999 that works is $(3 \times 3 \times 3)(37)$.

Thus the only divisor of 999 between 12 and 36 is 27 .

From this, we see that there are 27 cookies.

An ordinary cookie has $\frac{999}{27}=37$ chocolate chips, and the special cookie has 38 chocolate chips.

The required sum is $27+38=65$.

ANswer: (A)