Bunuel wrote:
The remainder when positive integer N is divided by 2 is 1, when divided by 3 is 2, when divided by 4 is 3, and when divided by 5 is 4. What is the sum of the digits of the least possible value of N?
(A) 11
(B) 13
(C) 14
(D) 16
(E) 17
Let’s use the last condition first: when N is divided by 5, the remainder is 4. The reason is: 5 is the largest divisor here, so if we list the numbers that satisfies this condition, the numbers will increase faster than those of the other three conditions. Recall that if the remainder is 4 when a number is divided by 5, the units digit must be either 4 or 9. However, it cannot be 4 since we are told that there is a remainder of 1 when the N is divided by 2. Now let’s list the numbers:
9, 19, 29, 39, 49, 59, 69, …
We can also omit the numbers 9, 39, 69, etc. since these numbers are divisible by 3 (we are given that N is not divisible by 3). So we are left with:
19, 29, 49, 59, 79, 89, …
We see that all of these numbers have a remainder of 1 when divided by 2, so let’s check division by 3 and 4:
19/3 = 6 R 1…. No
29/3 = 9 R 2…. Yes; 29/4 = 7 R 1 …. No
49/3 = 16 R 1 …. No
59/3 = 19 R 2…. Yes; 59/4 = 14 R …. Yes
Thus 59 is the least possible value of N and 5 + 9 = 14.
Alternate solution:
Notice that each remainder is the largest possible remainder of their respective divisor (for example, 3 is the largest possible remainder when a number is divided by 4). Therefore, if we can the find a number that has remainder of 0 when it’s divided by 2, 3, 4 and 5, and then subtract 1 from it, we will have a suitable value for N. Since we want the least possible value of N, we want to find the smallest positive integer that is divisible by 2, 3, 4 and 5. This number is, of course, the least common multiple of the 4 numbers and it is:
LCM(2, 3, 4, 5) = LCM(2, 3, 2^2, 5) = 2^2 x 3 x 5 = 60.
Since 60 is the LCM of (2, 3, 4, 5), 60 - 1 = 59 is the least possible value of N. Let’s verify it:
59/2 = 29 R 1; 59/3 = 19 R 2; 59/4 = 14 R 3 and 59/5 = 11 R 4.
Thus the sum of the digits of N = 5 + 9 = 14.
Answer: C