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# Quant Aptitude: Number System Solved Problems

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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.

Instructions
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 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
View Ans
 Answer – (D) Solution: Let the original number be 'a' Let the divisor be 'd' Let the quotient of the division of a by d be 'x' Therefore, we can write the relation as $a/d = x$ and the remainder is 24. i.e., $a = dx + 24$ When twice the original number is divided by d, 2a is divided by d. We know that $a = dx + 24$. Therefore, $2a = 2dx + 48$ The problem states that  $(2dx + 48)/d$ leaves a remainder of 11. 2dx is perfectly divisible by d and will therefore, not leave a remainder. The remainder of 11 was obtained by dividing 48 by d. When 48 is divided by 37, the remainder that one will obtain is 11. Hence, the divisor is 37.
 2 The product of 4 consecutive even numbers is always divisible by:
 A. 600 B. 768 C. 864 D. 384
View Ans
 Answer – (D) Solution: The product of 4 consecutive 4 numbers is always divisible by 4!. Since, we have 4 even numbers, we have an additional 2 available with each number. 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
View Ans
 Answer – (B) Solution: The marbles used to tile the floor are square marbles. Therefore, the length of the marble = width of the marble. As we have to use whole number of marbles, the side of the square should a factor of both 5 m 78 cm and 3m 74. And it should be the highest factor of 5 m 78 cm and 3m 74. $5 m 78 cm = 578 cm\text" and "3 m 74 cm = 374 cm.$ The HCF of 578 and 374 = 34. Hence, the side of the square is 34. The number of such square marbles required $= {578×374}/{34×34} = 17×11=$ 187 marbles
 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
View Ans
 Answer – (A) Solution: According to remainder theorem when ${f(x)}/{x+a}$, then the remainder is f(-a). In this case, as $x + 3$ divides $x^3 + 4x^2 - 7x + 12 – k$ perfectly (k being the number to be subtracted), the remainder is 0 when the value of x is substituted by -3. i.e., $(-3)^3 + 4(-3)^2 - 7(-3) + 12 - k = 0$ or $-27 + 36 + 21 + 12 = k$ or $k =$ 42
 5 Find the remainder when $2^89$ is divided by 89?
 A. 1 B. 2 C. 87 D. 88
View Ans
 Answer – (B) Solution: when we take successive powers of 2 and find their remainders, we get the following cyclic patterns of cycle length 11. viz $2,4,8,16,32,64,39,78,67,45, 1$ i.e $2^11$ leaves a remainder 1. Thus $2^89 = (2^11)^8 (2)$ leaves a remainder of 2.