# Numbers high level Exercise 2 - 1

1. 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 1000 for which ${P_n} + {S_n} = n$
a. 9
b. 100
c. 128
d. 145

2. 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

3. 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

4. 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

5.  By which number the expression $\dfrac{200!}{12^{100}}$ should be multiplied so that the given expression becomes an integer
a. 216
b. 200
c. 64
d. 9

6.  What is the maximum power of 3 in the expansion of 1! × 2! × 3! × . . . . × 100!
a. 11
b. 13
c. 14
d. 18