Number System Questions for CAT with Answers PDF

The units digit in this case would obviously be ‘0’ because the given expression has a pair of 2 and 5 in it’s prime factors.

The two numbers should be factors of 405. A factor search will yield the factors. (look only for 2 digit factors of 405 with sum of digits between 1 to 19).
Also 405 = 5 × 3⁴. Hence: 15 × 27
45 × 9 are the only two options. 
From these factors pairs only the second pair gives us the desired result. 
i.e. Number × sum of digits = 405.
Hence, the answer is 45.

Two more than half of 1/3rd of 96 = 18. Also since we are given  that the difference between the AM and GM is 18, it means that the GM must be an integer.
From amongst the options, only option (a) gives us a GM which is an integer.
Thus, checking for option 1, we get the GM = 7 and AM = 25.

For 381A to be divisible by 9, the sum of the digits 3 + 8 + 1 + A should be divisible by 9.
For that to happen A should be 6.
Option (d) is correct.

LCM of 5, 15 and 20 = 60.
HCF of 5, 15 and 20 = 5.
The required ratio is 60:5 = 12:1

Only the first option can be verified to be true in this case.
If A is even, 3A would always be divisible by 6 as it would be divisible by both 2 and 3.
Options b and c can be seen to be incorrect by assuming the value of A as 4.

B would necessarily be even- as the possible values of B for the three digit number 15B to be divisible by 6 are 0 and 6.
Also, the condition stated in option (c) is also seen to be true in this case — as both 0 and 6 are divisible by 6.
Thus, option (d) is correct.

The units digit would be given by 5 + 6 + 9 (numbers ending in 5 and 6 would always end in 5 and 6 irrespective of the power and 3⁵⁴ will give a units digit equivalent to 3⁴ⁿ⁺² which would give us a unit digit of 3² i.e.9).

The respective units digits for the six parts of the expression would be: 1 + 4 + 7 + 6 + 5 + 6 = 29 → required answer is 9.
Option (b) is correct.

The number of zeroes would be given by adding the quotients when we successively divide 1090 by 5 : 1090/5 + 218/5 + 43/5 + 8/5 = 218 + 43 + 8 + 1 = 270. Option (a) is correct.