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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

25. A 4 digit number is formed by repeating a 2 digit number such as 2525, 3232 etc. Any number of this form is exactly divisible by
 
  • A. 7
  • B. 11
  • C. 13
  • D. Smallest 3 digit prime number
Ans: D.
Sol.
Smallest 3 digit prime number is 101.
Clearly, 2525 = 25 x 101;
3232 = 32 x 101, etc.
∴ Each such number is divisible by 101.
 
 
26. The value of P, when 4864 x 9P2 is divisible by 12, is
 
  • A. 2
  • B. 5
  • C. 8
  • D. None of these
Ans: D.
Sol.
Since 4864 is divisible by 4, So 9P2 must be divisible by 3.
∴ (11 + 9) must be divisible by 3.
∴ Least value of P is 1.
 
 
27. The difference between two numbers is 1365. When the larger number is divided by the smaller one, the quotient is 6 and the remainder is 15. The smaller number is
 
  • A. 240
  • B. 270
  • C. 295
  • D. 360
Ans: B.
Sol.
Let the smaller number be x.
Then, larger number = (1365 + x).
∴ 1365 + x = 6x + 15
⇔ 5x = 1350
⇔ x = 270.
Hence, the required number is 270.