j_shreyans wrote:
The three-digit positive integer n can be written as ABC, in which A, B, and C stand for the unknown digits of n. What is the remainder when n is divided by 37?
(1) A + B/10 + C/100 = B + C/10 + A/100
(2) A + B/10 + C/100 = C + A/10 + B/100
IMPORTANT point: The VALUE of a 3-digit integer xyz is 100x + 10y + z
Example: 723 = (7)(100) + (2)(10) + 3
Target question: What is the remainder when n is divided by 37?
Statement 1: A + B/10 + C/100 = B + C/10 + A/100
Multiply both sides by 100 to get: 100A + 10B + C = 100B + 10C + A
On the left-hand side, C represents the units digit of the sum, and on the right-hand side, A represents the units digit of the sum.
Since both sums (left and right) are equal, we can conclude that
C = A
Likewise, on the left-hand side, B represents the tens digit of that sum, and on the right-hand side, C represents the tens digit of that sum.
So, we can conclude that
B = C
If
C = A and
B = C, we can conclude that
A = B = C
In other words, all 3 digits of the number ABC ARE EQUAL.
This means that n could be 111, 222, 333, ...,888, or 999
Notice that all of these possible n-values are divisible by 111.
Since 111 itself is divisible by 37, ALL possible values of n (111, 222, 333, etc) are divisible by 37.
So,
when n is divided by 37, the remainder must be zero
Since we can answer the
target question with certainty, statement 1 is SUFFICIENT
Statement 2: A + B/10 + C/100 = C + A/10 + B/100
Multiply both sides by 100 to get: 100A + 10B + C = 100C + 10A + B
Using the same logic that we used for statement 1, we can see that
A = B = C
So, once again, we can conclude that
when n is divided by 37, the remainder must be zero
Since we can answer the
target question with certainty, statement 2 is SUFFICIENT
Answer =
D
Cheers,
Brent
