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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

4. 5978 + 6134 + 7014 = ?
 
  • A. 16226
  • B. 19126
  • C. 19216
  • D. 19226
Ans: B.
Sol.
5978 + 6134 + 7014 = 19126.
 
5. How many numbers between 200 and 600 are divisible by 4, 5 and 6?
 
  • A. 5
  • B. 6
  • C. 8
  • D. 10
Ans: B.
Sol.
Every such numbers must be divisible by L.C.M. of 4, 5, 6, i.e. 60.
Such numbers are 240, 300, 360, 420, 480, 540.
Clearly, there are 6 such numbers.
 
 
6. The least number by which 72 must be multiplied in order to produce a multiple of 112, is
 
  • A. 6
  • B. 12
  • C. 14
  • D. 18
Ans: C.
Sol.
Required number is divisible by 72 as well as by 112, If it is divisible by their LCM, Which is 1008.
Now, 1008 when divided by 72, gives quotient = 14.
Therefore, required number = 14.