n= 3-digit positive integer = ABC where
A= hundredth value of n
B= Tenth value of n
C= Unit value of n
Question=> What is the remainder when n is divided by 37?
i.e Find ABC and 37
$$Statement\ 1:\ A+\frac{B}{10}+\frac{C}{100}=B+\frac{C}{10}+\frac{A}{100}$$
$$\frac{\left(100A+10B+C\right)}{100}=\frac{\left(100B+10C+A\right)}{100}$$
$$Remember\ that\ A=hundredth\ value\ of\ n$$
$$Therefore,\ 100A=A;\ 10B=B;\ C=C.$$
$$So,\ \frac{A+B+C}{100}=\frac{B+C+A}{100}$$
Divide through by 100, we have (A+B+C) = (B+C+A)
If (A+B+C) = (B+C+A), then they must have the same hundredth digit A=B; the same tenth B=C an dthe same unit C=A.
Therefore, A=B=C and n= ABC = AAA, this means that A, B, and C have the same value.
If ABC=111 then 111/37 = 3 remainder 0.
If ABC=222 then 222/37 = 6 remainder 0.
If ABC=999 then 999/37 = 27 remainder 0.
Therefore, ABC and 37 = 0. Statement 1 is SUFFICIENT.
$$Statement\ 2:\ A+\frac{B}{10}+\frac{C}{100}=C+\frac{A}{10}+\frac{B}{100}$$
This is almost the same as statement 1 and we will have A+B+C = C+A+B and A=C; B=A; C=B.
Therefore, A=C=B=A and n=ABC=AAA which means; same value for the 3-digits ABC and 37 = 0.
Statement 2 is also SUFFICIENT.
Since each statement alone is SUFFICIENT, answer = option D
