SOLVED EXAMPLES

 SOLVED EXAMPLES

 Ex. 1. 9587? = 7429 – 4358.

Sol let 9587 – X = 7429 – 4358. Then,

9587 – X = 3071 → X = 95 87 – 3071 = 6516

Ex 2. 5793405 x 9999

Sol. 5793405 x 9999= 5793405 x (10000 – 1)

              =57934050000 – 5793405

              = 57928256595

Ex. 3. 839478 x 625 = 839478 x 5²

      Sol        = 839478 x 625 = 839478 x 5⁴

                  = 839478 x (10/2)⁴ = 839478x 10⁴/2⁵

                  =8394780000/16 = 524673750

Ex.4. 976x 236 +976 x 763=?

Sol. Using distributive law, we get:

976 x 237 + 976+ 763 = 976 x (237 +763)

=976 x 1000 = 976000.

Ex 5. 986 x 307 – 986 x 207 =?

Sol. By distributive law, we get.

986 x 307 -986x 207 =986 x (307 – 207)

= 986 x 100 = 98600

2560000                 +  49              +22400

1582449

Ex 6. 1607 x 1607 =?

Sol. 1607 x 1607 = (1607)²

  = (1600 + 7)²= (1600)² + 7² + 2 x 1600 x 7

  = 2560000 + 49 + 22400 = 2582449.

  

1960016       -11200

1948816

 Ex. 7. 1396 x 1396 =?

Sol. 1396 x 1396 = (1396)²

      = (1400 – 4)² = (1400)² + 4² - 2 x 14000x4

     = 1960000 + 49- 11200 = 1948816

Ex. 8. (475 x 475 + 125 x 125) = ?

Sol. We have (a² + b²) = ½ [(a + b)² + (a – b )²]

      (475)²+ (125)² = 1/2 [(475 + 125)²(475 – 125)²]

                                =1/2 x 482500 =241250.

[(a² -b²) = (a + b) (a – B)]

Ex. 9. (9 x 96 – 204 x 204) = ?

Sol. 796 + 9 – 204 x 204 = (796)² - (204)²

  = (796 + 204) (796 – 204)

= (1000 x 593) = 592000.

Ex 10. (387x 387 + 113 x113 + 2 x 387 x 113)=?

Sol. Given exp. = (387)² + (113)² +2x 387 + 113

(a² +b²+ 2ab), where a = 387 and b = 113

= (a +b)² = (387+ 113)²= (500)² = 250000

Ex. 11. (87x 87 + 61- 2 x 87 x 61)= ?

Sol. Given exp. = (87)² + (61)² -2 x 87 x 61

= (a² + b² - 2ab ), where a = a =87 and b=61

= (a - b) ²= (87 -61)² = (26)² = (20 + 6)²

= (20)² + 6²+ 2 x20x6 = (400 + 36 – 240)

= (436 – 240) = 196.

Ex. 12. Find the least value of * for which 5967 * 13 is divisible by 3.

Sol let the required value be X. then,

(5 + 9+6+7+X+1+3)=(31+X) is divisible by 3.

Least value of X 2.

 Ex. 13 Find the least value of * for which 7 * 5462 is divisible by 9.

Sol. Let the required value be X. then.

(7 + X + 5 + 4 + 6 + 2) =(24 +  X) is visible by 9.

Ex. 14. Find the least value of * for which 4832* is divisible by 11.

sol. (sum of digits at odd places) -(sum of digits as even places)

- (8+X+3+4)- (1+2+8) =(4+X), which should be divisible by 11.

X=7

Ex. 15. show that 52563744 is divisible by 24. 

Sol. 24 =3 x 8, where 3 and 8 are co-prime.

Sum for digits =36, which is divisible by 3.

So, the given number is divisible by 3.

 The number formed by last 3 digits = 744, which is divisible by 8.

So, the given number is divisible by 8.

Hence, it is divisible by (3.x8), i.e., 24.