•   2 , 3 , 4 and 8 can be rewritten as 2 , 3 , 2 and 2 .

•   It will be 2⁴⁰ , 3²¹, 2³⁶ , 2³⁶

•   Third and fourth numbers are equal thus they cant be smallest.

•   First number is greater than these two numbers thus that also can not be smallest.

•   Thus smallest number is 3²¹

•   Prime factorize 1261 = 13 × 97.

•   13 and 97 both are prime, so can’t be further factorized.

•   Both 13 and 97 are available in the given product.

•   Thus, 1261 will divide the given product completely.

•   So the remainder will be 0.

•   15 × 14 × 13 × ….. × 3 × 2 × 1 = 15!

•   We have to find maximum power of 3 in 15!

•   That will be in 15/3 (= 5) + 5/3 (= 1) = 6 [integer value of  15/3 integer value of 15/9  integer value of 15/27 = 5 + 1 + 0 = 6 ]

•   Otherwise, in the multiplication of 15 × 14… 2 × 1, 3 is present 15 (one time), 12 (one time), 9 (two times), 6(one time) and 3 (one time). So, total 1 + 1 + 2 + 1 + 1 = 6 times.

•   Thus, maximum power of 3 will be 6 in 15!

•   Since, unit place of the power of 3 repeats after every 4 steps (i.e., it has a cyclicity of 4).

•   Now, dividing 2019 by 4 we get a remainder of 3.

•   Hence, 3^2019 will have the same last digit as that of 33, i.e., 7.

•   (3^3)/10 = 27/10 Hence, the remainder will be 7.

•   Let's try to divide 3798125 by 7. It gives 2 as remainder.

•   Thus, remaining number is 2P369.

•   Taking P = 6 we get 26369/7 = 3767

•   Thus, P = 6. Thus, option (b) is correct answer.

•   Following are the integers from 700 to 1000.

•   Where sum of their digits is 10.

•   703, 712, 721, 730 = 4

•   802, 811, 820 = 3

•   901, 910 = 2

•   In total there are 9 such integers between 700 and 1000.

•   3P + 4P + PP + PP = RQ2

•   Or 30 + P + 40 + P + 10P + P + 10P + P = 100R + 10 Q + 2

•   Or 24P + 70 = 100R + 10 Q + 2

•   Or 20P + 70 + 4P = 100R + 10 Q + 2.

•   The unit digit of the resultant is 2.

•   It will be obtained when 4 is multiplied by P.

•   So, P must be 3, or 8. If P = 3, then: 24P + 70 = 24 × 3 + 70 = 72 + 70 = 142.

•   If P = 8, then: 24P + 70 = 24 × 8 + 70 = 192 + 70 = 262.

•   Arithmetic sum of 142 and 262 = (142 + 262)/2 = 202.

•   Here if we take Q = 5 only then after multiplication by 3 the units place will also be 5.    

•   P5    × 3  = R55

•   Again, putting P = 8 we get 85 × 3 = 255 ⇒ R = 2, P = 8, Q = 5.

•   For any other value of P i.e. less than 8 overall multiplication value does not have 55 at the end.

•   Hence (P+R) /Q = (8 + 2)/ 5 = 2.

•   Hence, option 2 is correct.

•  (½)-6 = 26 = 64

•  (1/4)-3 = 43 = 64

•  (1/3)-4 = 34 = 81

•  (1/6)-2 = 62 = 36

•   Pillar X: In one chance, A climbs on X by 6 cm but slips down 1 cm. Hence, in one chance A climbs 6 – 1 = 5 cm.

•   Height attained after 39 chances = 39 × 5 = 195 cm

•   Thereafter in the 40th and last chance it will climb 6 cm to reach the top. So, height of pillar X = 195 + 6 = 201 cm

•   Pillar Y: In one chance, B climbs on Y by 7 cm but slips down 3 cm. Hence, in one chance B climbs 7 – 3 = 4 cm.

•   Height attained after 39 chances = 39 × 4 = 156 cm

•   Thereafter in the 40th and last chance it will climb 7 cm to reach the top. So, height of pillar Y = 156 + 7 = 163 cm

•   Pillar Z: In one chance, C climbs on Z by 6.5 cm but slips down 2 cm. Hence, in one chance C climbs 6.5 – 2 = 4.5 cm.

•   Height attained after 39 chances = 39 × 4.5 = 175.5 cm

•   Thereafter in the 40th and last chance it will climb 6.5 cm to reach the top. So, height of pillar Z = 175.5 + 6.5 = 182 cm

•   So, height of the shortest pillar (i.e. pillar Y) = 163 cm

•   Arithmetic mean is also known as average. It is the sum of all the numbers divided by the number of numbers.

•   Median is the middle value in the list of numbers written in ascending or descending order.

•   Mode is the most frequently occurring value in the data set.

•   A Normal Distribution is perfectly symmetrical (not skewed). It means Mean = Mode = Median.

•   A distribution is skewed if one tail is longer than another. These distributions are sometimes called asymmetric distributions. These are of two types:

1. If Mode < Mean, then the data is skewed to the right.

2. If Mode > Mean, then the data is skewed to the left.

•   We have printed from page 1 to 150.

•   Total digits from pages 1 to 9 = 9.

•   Total digits from pages 10 to 99 = 90 x 2 = 180.

•   Total digits from pages 100 to 150 = 51 x 3 = 153.

•   Let the number of bees = x

•   Then the number of flowes = (x - 1)

•   If the number of bees sitting on each flower is 2, number of flowers is (x/2 + 1).

⇒ (x/2 + 1) = (x - 1)

⇒  x - x/2 = 2

⇒ x/2 = 2

⇒ x = 4

•   Therefore, number of bees is 4 and the number of flowers is (4 - 1) i.e. 3.

•   Person is standing on first step and middle step is 4 step ahead.

•   Therefore middle step is the 5th step. Now top step will also be 4 step ahead of the middle step.

•   Top step = 5 + 4 = 9 steps

•   Total no. of steps in the ladder = 9.