(16) Find  the  unit’s  digit  in  the  product  (5327)¹⁵³ × (143)⁷².

Soln. First of  all divide  the index  by  4 and  find  out  its  remainder.

   ( Replace the remainder 4 by 1 and others  remain  as  they  are. )

i.e.    (5327)¹⁵³  ×  (143)⁷²    =    7¹ × 3⁴          (we need unit’s digit only )

=    7 × 3² × 3²

=    7 × 9 × 9      (we need unit’s digit only )

=    7 × 81

=    7                   (we need unit’s digit only )

(17) Find the unit’s digit in (784)¹⁰² + (784)¹⁰³.

Soln. First of  all divide  the index  by  4 and  find  out  its  remainder.

( Replace  the  remainder  4  by  1  and  others  remain  as  they  are. )

i.e.      (784)¹⁰²  +  (784)¹⁰³     =   4²  +  4³           (we need unit’s digit only )

=   16   +   64

=    6   +   4          (we need unit’s digit only )

=    10

=     0                   (we need unit’s digit only )

(18) Find  the  remainder  when  2³⁵  is  divided  by   5.

Soln. Unit digit of    2³⁵  =   2³   =   8

Now, 8 when divided by  5,  gives 3 as remainder.

Hence,  2³⁵  when  divided  by  5,  gives  3  as  remainder.

(19) What least value must be assigned to *  so that the number 383*1386  is divisible by 9 ?

Soln. Sum of  digits  =  ( 3  +  8  +  3 +  *  +  1  +  3  +  8  +  6 )  =  (32 + *).

For  ( 32 +  * )  to  be  divisible  by   9,  *  must  be  replaced  by  4.

Hence,  the  digit  in  place  of    *  must  be  4.

(20) Which digits should come in place of  *  and  #  if the number  35264*#  is divisible by both  8  and  5 ?

Soln. Since the given number is divisible by 5, so 0 or 5 must come in place of  # . But, a number ending with  5  is never divisible by  8. So,  0  will replace  #.

Now,  the number  formed  by  the last three digits is  4*0,  which becomes divisible by 8,  if  *  is replaced by  0   or   4.

Hence, digits in place of  *  =  0  or   4  and  #   = 0.

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