Bunuel wrote:
When p is divided by 7, the remainder is 4. When p is divided by 4, the remainder is 1. How many different values of p are less than 120?
(A) 2
(B) 4
(C) 5
(D) 11
(E) 18
We have the following two equations:
p = 7Q + 4 where Q is a whole number.
So, p can be values such as 4, 11, 18, 25, …, 7(16) + 4 = 116.
and
p = 4S + 1 where S is a whole number.
So, p can be values such as 1, 5, 13, 14, 21, 25, …, 4(29) + 1 = 117.
We see that 25 is the first matching value of p. Since the LCM of 7 and 4 is 28, the next will be 25 + 28 = 53, and the next will be 53 + 28 = 81, and the final matching value will be 81 + 28 = 109, so there are a total of 4 values of p less than 120 that satisfy the two criteria.
Alternate Solution:
We have
p = 7Q + 4
for some Q and
p = 4S + 1
for some S.
Then, p + 3 = 7Q + 7 = 4S + 4 is divisible by both 7 and 4. Let’s list the numbers less than 120 that are divisible by 7 and 4, i.e., by 28: 0, 28, 56, 84, 112. However, these are values for p + 3; thus, the corresponding values for p are -3, 25, 53, 81, and 109. p can take any value in this list except for -3; therefore, there are four values in total.
Answer: B
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