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Examples 11 to 15 of Fundamental(Numbers)

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(11) What least number must be added to 16759 to get a number exactly divisible by 37 ?

Soln. On dividing 16759 by 37, the remainder is 35.

$\therefore$ The number to be added = ( 37 – 35 ) = 2.

(12) Simplify : $\frac{1}{1+\frac{1}{2+\frac{1}{3+\frac{1}{4+\frac{1}{3}}}}}$

Soln. $\frac{1}{1+\frac{1}{2+\frac{1}{3+\frac{1}{4+\frac{1}{3}}}}}$

= $\frac{1}{1+\frac{1}{2+\frac{1}{3+\frac{1}{\frac{13}{3}}}}}$

= $\frac{1}{1+\frac{1}{2+\frac{1}{3+\frac{3}{13}}}}$

= $\frac{1}{1+\frac{1}{2+\frac{13}{42}}}$

= $\frac{1}{1+\frac{42}{97}}$

= $\frac{97}{139}$

(13) Simplify : $\frac{1}{2+\frac{1}{2+\frac{1}{3-\frac{1}{3}}}}$

Soln. Given Exp. = $\frac{1}{2+\frac{1}{2+\frac{1}{\frac{8}{3}}}}$

= $\frac{1}{2+\frac{1}{2+\frac{3}{8}}}$

= $\frac{1}{2+\frac{1}{\frac{19}{8}}}$

= $\frac{1}{2+\frac{8}{19}}$

= $\frac{1}{\frac{46}{19}}$

= $\frac{19}{46}.$

(14) Find a + b, if 1938a5b is divisible by 33 ( i.e. by 3 and 11 both ).

Soln. Sum of digits = 1 + 9 + 3 + 8 + a + 5 + b = 26 + a + b is divisible by 3

∴ a + b could be 1, 4, 7, 10, 13 or 16

Also ( 1 + 3 + a + b ) – ( 9 + 8 + 5 ) = 0 or multiple of 11.

∴ ( 4 + a + b ) – ( 22 ) = a + b – 18 = 0 or multiple of 11. ( i.e. ±11, ±22, ±33 )

∴ a + b = 7 ( by substituting the different values of a + b in above equation ).

(15) What could be the maximum value of Q in the following equation ?

7P9 + 3R8 + 2Q7 = 1314

Soln. We may analyze the given equation as shown :

Clearly, 2 + P + R + Q = 11.

So, the maximum value of Q can be

(11 – 2) =. 9 ( when P = 0, R = 0 ).