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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

7. A number when divided by 6 leaves a remainder 3. When the square of the same number is divided by 6, the remainder is
 
  • A. 0
  • B. 1
  • C. 2
  • D. 3
Ans: D.
Sol.
Let x = 6q + 3.
Then x2 = (6q + 3)2
= 36q2 + 36q + 9 = 6(6q2 + 6q + 1) + 3.
So, when x2 is divided by 6, remainder = 3.
 
 
8. The difference between the place values of 7 and 3 in the number 527435 is
 
  • A. 4
  • B. 5
  • C. 45
  • D. 6970
Ans: D.
Sol.
= (Place value of 7) – (place Value of 3)
= (700 - 30)
= 6970.
 
 
9. A positive integer, which when added to 1000, gives a sum which is greater than when it is multiplied by 1000. This positive integer is
 
  • A. 1
  • B. 3
  • C. 5
  • D. 7
Ans: A.
Sol.
(1000 + N) > (1000N), Clearly N = 1.