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This subsection of Aptitude Test Solved Problems is on "Number System and Number Theory". These moderately difficult questions with detailed solutions on Number system are helpful for those who are preparing for competitive exams like MAT, SNAP, XAT, CAT, TISS, GATE aptitude, GMAT, GRE etc.

1. 
A number when divided by a divisor leaves a remainder of 24. When twice the original number is divided by the same divisor, the remainder is 11. What is the value of the divisor? 
A. 
13 

B. 
59 

C. 
35 

D. 
37 

Answer – (D) Solution: Let the original number be 'a' 
2. 
The product of 4 consecutive even numbers is always divisible by: 
A. 
600 

B. 
768 

C. 
864 

D. 
384 

Answer – (D) Solution: The product of 4 consecutive 4 numbers is always divisible by 4!. It is always divisible by $(2^4)4! = 16(24) =$ 384. 
3. 
What is the minimum number of square marbles required to tile a floor of length 5 metres 78 cm and width 3 metres 74 cm? 
A. 
176 

B. 
187 

C. 
540 

D. 
748 

Answer – (B) Solution: The marbles used to tile the floor are square marbles. Therefore, the length of the marble = width of the marble. 
4. 
What number should be subtracted from $x^3+ 4x^2 7x + 12$ if it is to be perfectly divisible by x + 3? 
A. 
42 

B. 
39 

C. 
13 

D. 
None of these 

Answer – (A) Solution: According to remainder theorem when ${f(x)}/{x+a}$, then the remainder is f(a). 
5. 
Find the remainder when $2^89$ is divided by 89? 
A. 
1 

B. 
2 

C. 
87 

D. 
88 

Answer – (B) Solution: when we take successive powers of 2 and find their remainders, we get the following cyclic patterns of cycle length 11. Thus $2^89 = (2^11)^8 (2)$ leaves a remainder of 2. 
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