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## 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. 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. 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

19. If x is a whole number, then x2(x2 - 1) is always divisible by

• A. 12
• B. 24
• C. 12-x
• D. multiple of 12
 Ans: A. Sol. Putting x = 2, we get 22 (22 - 1) = 12. So, x2(x2 - 1) is always divisible by 12.

20. The largest number that exactly divides each number of the sequence
(15 - 1), (25 - 2), (35 - 3), ...., (n5 - n), ...... is

• A. 1
• B. 15
• C. 30
• D. 120
 Ans: C. Sol. Required number = (25 - 2) = (32 - 2) = 30.

21. The digit in units place of the product 81 x 82 x ..... x 89 is

• A. 0
• B. 2
• C. 6
• D. 8
 Ans: A. Sol. Required digit = unit digit in (1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9) = 0.