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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

22. 325325 is a six digit number. It is divisible by
 
  • A. 7 only
  • B. 11 only
  • C. 13 only
  • D. all 7,11 and 13
Ans: D.
Sol.
Clearly, 325325 is divisible by all 7, 11 and 13.
 
 
23. When the sum of two numbers is multiplied by 5, the product is divisible by 15. Which one of the following pairs of numbers satisfies the above condition?
 
  • A. 240, 355
  • B. 250, 341
  • C. 245, 342
  • D. 250, 261
Ans: B.
Sol.
Clearly, 5 x (sum of numbers) is divisible by 15.
∴ Sum of numbers must be divisible by 3.
Now, (250+341) = 591 is divisible by 3. So, required pair is 250, 341.
 
 
24. 5*2 is a three digit number with * is a missing digit. If the number is divisible by 6, the missing digit is
 
  • A. 0
  • B. 2
  • C. 5
  • D. 7
Ans: B.
Sol.
Let the number be 5x2. Clearly, it is divisible by 2.
Now, 5 + x + 2 = (7 + x) must be divisible by 3.
So, x = 2.