# Number System Digit problems

Number system digit problems are very important type questions.  A sample problem goes like this:
"A two digit number when 18 added becomes another two digit number with reversed digits.  How many such two digit numbers are possible?"
Take a number 256.  In this number 6 is in units place, 5 is in tenth's place, 2 is in 100th place.  In decimal system, the place value increases by 10 times while going to the left.
So 256 can be written as 100 × 2  +  10 × 5 + 1 × 6
Similarly a four digit number abcd can be written as 1000 × a +  100 × b + 10 × c + 1 × d = 1000a + 100b + 10c + d

Very important note: When you represent a number in this format, Left most digit can take any value from 1 to 9 but not zero.   Remaining digits can take any value from 0 to 9.

## Exercise

1. A two digit number when 18 added becomes another two digit number with reversed digits. How many such two digit numbers are possible?
a. 2
b. 3
c. 7
d. 8

2. For a positive integer n, let $${P_n}$$ denote the product of the digits of $n$ and $${S_n}$$ denote the sum of the digits of $n$. The number of integers between 10 and 100 for which $${P_n} + {S_n} = n$$
a. 2
b. 3
c. 9
d. 12

3. 3. Ray writes a two digit number. He sees that the number exceeds 4 times the sum of its digits by 3. If the number is increased by 18, the result is the same as the number formed by reversing the digits. Find the sum of the digits of the number.
a. 8
b. 9
c. 10
d. 12

4. The digits of a three-digit number A are written in the reverse order to form another three-digit number B. If B>A and B-A is perfectly divisible by 7, then which of the following is necessarily true?
a. 100 < A < 299
b. 106 < A < 305
c. 112 < A < 311
d. 118 < A < 317

5. M = abc is a three digit number and N = cba, if M > N and M - N + 396c = 990. Then how many values of M are more than 300.
a. 20
b. 30
c. 40
d. 200

6. Consider four digit numbers for which the first two digits are equal and the last two digits are also equal. How many such numbers are perfect square?
a. 2
b. 4
c. 0
d. 1