This EduRev document offers 10 Multiple Choice Questions (MCQs) from the topic Number System (Level - 2). These questions are of Level - 2 difficulty and will assist you in the preparation of CAT & other MBA exams. You can practice/attempt these CAT Multiple Choice Questions (MCQs) and check the explanations for a better understanding of the topic.
Question for Practice Questions Level 2: Number System - 1
Try yourself:How many digits are required to number a book containing 200 pages?
Explanation
Total number of pages = 200
Number of digits required to number the book from page 1 to 9 = 9
Number of digits required to number the book from page 10 to 99 = 2 × 90 = 180
Number of digits required to number the book from page 100 to 200 = 3 × 101 = 303
Therefore, total number of digits = 303 + 180 + 9 = 492
Question for Practice Questions Level 2: Number System - 1
Try yourself:If X381 is divisible by 11 and 381Y is divisible by 9, find the smallest values of X and Y.
Explanation
(i) X381 is the number given to us. If this number is divisible by 11, then the divisibility rule for 11 must be satisfied.
i.e. (X + 8) - (3 + 1) = (X + 4) is multiple of 11 and the smallest value of X for which X + 4 is divisible by 11 is 7. Hence, X = 7
(ii) 381Y is the number given to us. If this number is divisible by 9, then the divisibility rule for 9 must be satisfied.
i.e. (3 + 8 + 1 + Y) = Y + 12 must be divisible by 9. And smallest value of Y for which Y + 12 is divisible by 9 is 6. Hence, Y = 6
Question for Practice Questions Level 2: Number System - 1
Try yourself:If the sum of any two positive quantities is constant, then their product is maximum when the two quantities
Explanation
If the sum of two quantities is constant, then their product is maximum when they are equal.
Alternative Method:
Let X + Y = K
(X + Y)^{2} - (X - Y)^{2} = 4XY
K^{2} - (X - Y)^{2} = 4XY
⇒ 4XY < K^{2} (except when X - Y = 0)
Or,
X = Y
⇒ 4XY is maximum when X = Y.
⇒ XY is maximum when X = Y = (K/2)
Thus, the maximum value of XY = (K^{2}/4)
Question for Practice Questions Level 2: Number System - 1
Try yourself:If p, q, r, s and t are integers, which of the following must be even for the expression p^{2}{q^{3}(r - s) + t} to be an even number?
Explanation
The expression is of the form p^{2}X, where X = {q^{3}(r - s) + t}.
So, whenever p is even, the product will be even, irrespective of whether the second number X is odd or even. So, p must be even.
Question for Practice Questions Level 2: Number System - 1
Try yourself:If P, Q and R are three consecutive integers, then which of the following is always true?
Explanation
The sum of two extreme numbers (whether both odd/even) and twice the 3^{rd} number is always even.
Therefore, their sum is always even.
Alternative:
If P and R are odd and Q is even, then P + R is even and 2Q is even.
P + 2Q + R is even
If P and R is even and Q is odd, then P + R is even and 2Q is even.
P + 2Q + R is even.
Question for Practice Questions Level 2: Number System - 1
Try yourself:Two positive whole numbers are such that the sum of the first and twice the second is 8 and the difference between the numbers is 2. The numbers are:
Explanation
Let the first number be x and the second number be y.
Then, x + 2y = 8 and x - y = 2
x + 2(x - 2) = 8
x + 2x - 4 = 8
3x = 12
x = 4
Since x - y = 2, this implies that y = 2.
Question for Practice Questions Level 2: Number System - 1
Try yourself:A lady says that her age will be 18 years if only weekend days are counted. What is her actual age?
Explanation
If only weekend days are counted, only 2/7 portion of the whole week is calculated to arrive at her age.
So, 18 years of age = 2/7 (Actual age) ⇒ Actual age = 63 years
Question for Practice Questions Level 2: Number System - 1
Try yourself:Find the last digit of (173)^{99}.
Explanation
3^{1} = 3, 3^{2} = 9, 3^{3} = 27, 3^{4} = 81 and 3^{5} = 81 × 3 = 243.
Use the concept of power cycle (units digit repeats after the power of 4 and then after all powers that are multiples of 4).
Now, 99 = 24 × 4 + 3
Thus, 99 has 24 complete cycles of 4 and 3 is left as the remainder.
So, units digit of (173)^{99} will be units digit of 3^{3}, which is 7.
Question for Practice Questions Level 2: Number System - 1
Try yourself:What is the right most non-zero digit of 13780000^{13780000}?
Explanation
The required answer is nothing but the last digit of 8^{13780000}.
We know that, the cyclicity of 8 is 4.
As 13780000 is divisible by 4 (cyclicity of 8).
That is, the last digit of 8^{4} = 6
Question for Practice Questions Level 2: Number System - 1
Try yourself:The highest factor of 1573, except itself, is
Explanation
1573 = 11 × 143
= 11 × 11 × 13
So, highest factor = 11 × 13 = 143