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Numbers

Aptitude

Numbers -> IMPORTANT FACTS AND FORMULAE

1. Natural Numbers :
Counting numbers 1, 2, 3, 4, 5, .. are called natural numbers.
II. Whole Numbers :
All counting numbers together with zero form the set of whole numbers. Thus,
I. 0 is the only whole number which is not a natural number.
II. Every natural number is a whole number.
III.Some Important Formulae :
I. ( 1 + 2 + 3 + .....+ n) = n (n + 1 ) / 2
II. (1 2 + 22 + 32 + ..... + n2) = n ( n + 1 ) (2n + 1) / 6
III. (1 3 + 23 + 33 + ..... + n3) = n2 (n + 1)2 / 4

Numbers -> SOLVED EXAMPLES

1. 217 × 217 + 183 × 183 = ?
  Sol. Let the speed of the motorboat in still water be x kmph. Then,
Speed downstream = (x + 2) kmph; Speed upstream = (x - 2) kmph.
∴ 6/x+2 + 6/x-2 = 33/60
⇔ 11x² - 240x - 44 = 0
⇔ 11x² - 242x + 2x - 44 = 0
⇔ (x - 22) (11x + 2) = 0 ⇔ x = 22.
Hence, speed of motorboat in strill water = 22 kmph.
2. A man can row 18 kmph in still water. It takes him trice as long to row up as to row down the river. Find the rate of stream.
  Sol. Let man’s rate upsream be x kmph. Then, his rate downstream = 3x kmph.
∴ Rate in still water = 1/2(3x + x) kmph = 2x kmph.
So, 2x = 18 or x = 9.
∴ Rate upstream = 9 km/hr, Rate downstream = 27 km/hr.
Hence, rate of stream = 1/2(27 - 9) km/hr = 9 km/hr.
3. A man can row upstream at 7 kmph and downstream at 10 kmph. Find man’s rate in still water and the rate of current.
  Sol. Rate in still water = 1/2 (10 + 7) km/hr = 8.5 km/hr,
Rate of current = 1/2 (10 - 7) km/hr = 1.5 km/hr.

Numbers -> Exercise

28. What least value must be assigned to * so that the number 63576*2 is divisible by 8?
 
  • A. 1
  • B. 2
  • C. 3
  • D. 4
Ans: C.
Sol.
The given number is divisible by 8, if the number 6x2 is divisible by 8.
Clearly, the least value of x is 3.
 
 
29. A number is multiplied by 11 and 11 is added to the product. If the resulting number is divisible by 13, the smallest original number is
 
  • A. 6
  • B. 12
  • C. 22
  • D. 52
Ans: B.
Sol.
Let the required number be x.
Then, (11x + 11) = 11(x + 1) is divisible by 13. So, x = 12.
 
 
30. The largest number that exactly divides each number of the sequence
(15 - 1), (25 - 2), (35 - 3), ....., (nn - n), ....... is
 
  • A. 5
  • B. 15
  • C. 30
  • D. 45
Ans: C.
Sol.
Required number = (25 - 2)
= (32 - 2) = 30.