Number System Questions for CAT with Answers PDF

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

(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

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
K² - (X - Y)² = 4XY
⇒ 4XY < K² (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 = (K²/4)

The expression is of the form p²X, where X = {q³(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.

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.

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.

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

3¹ = 3, 3² = 9, 3³ = 27, 3⁴ = 81 and 3⁵ = 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 3³, which is 7.

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 8⁴ = 6

1573 = 11 × 143
= 11 × 11 × 13
So, highest factor = 11 × 13 = 143