Number System - Study Mode

[#286] Find the remainder when 67 107 is divided by 7.
Correct Answer

(C) 1

Explanation

Solution: $$eqalign{
& frac{{{{67}^{107}}}}{7} cr
& Or,,frac{{{{left( {7 imes 9 + 4}
ight)}^{107}}}}{7} cr
& { ext{The}},{ ext{remainder}},{ ext{will}},{ ext{be}},{ ext{same}},{ ext{as}} cr
& frac{{{4^{107}}}}{7} cr
& Or,,frac{{{4^3}}}{7} cr
& Or,,frac{{64}}{7} cr
& { ext{Required}},{ ext{Remainder}} = 1 cr} $$

[#287] Find the remainder when 54 124 is divided by 17.
Correct Answer

(B) 13

Explanation

Solution: $$eqalign{
& frac{{{{54}^{124}}}}{{17}} cr
& Or,,frac{{{{left( {17 imes 3 + 3}
ight)}^{124}}}}{{17}} cr
& { ext{The}},{ ext{remainder}},{ ext{would}},{ ext{be}},{ ext{same}},{ ext{as}}, cr
& frac{{{3^{124}}}}{{17}} cr
& Or,,frac{{{3^{4 imes 31}}}}{{17}} cr
& { ext{Remainder}},{ ext{will}},{ ext{be}},{ ext{same}},{ ext{as}}, cr
& frac{{{3^4}}}{{17}} cr
& Or,,frac{{81}}{{17}} cr
& ext{Remainder} = 13 cr} $$

[#288] Find unit digit of product (173) 45 × (152) 77 × (777) 999 .
Correct Answer

(C) 8

Explanation

Solution: To find unit digit of a number or an Expression, We have to divide the number or expression by 10 and the remainder obtained by this operation would be the required unit digit. $$eqalign{
& frac{{ {{{left( {173}
ight)}^{45}} imes {{left( {152}
ight)}^{77}} imes {{left( {777}
ight)}^{999}}} }}{{10}} cr
& { ext{Remainder}},{ ext{would}},{ ext{be}},{ ext{same}},{ ext{as}}, cr
& frac{{ {{3^{45}} imes {2^{77}} imes {7^{999}}} }}{{10}} cr
& Or,,frac{{ {3 imes 2 imes {7^3}} }}{{10}} cr
& Or,,frac{{ {6 imes 343} }}{{10}} cr
& { ext{Remainder}},{ ext{would}},{ ext{be}},{ ext{same}},{ ext{as}},frac{{ {6 imes 3} }}{{10}} cr }$$ Thus, Required remainder and unit digit will be 8.

[#289] The square of a number greater than 1000 that is not divisible by three, when divided by three, leaves a remainder of
Correct Answer

(A) 1 always

Explanation

Solution: In such cases remainder will always be 1.

[#290] If 146 Is divisible by 5 n , and then find the maximum value of n.
Correct Answer

(B) 35

Explanation

Solution: $$eqalign{
& { ext{Required}},{ ext{answer}}, cr
& = {frac{{146}}{5}} + {frac{{146}}{{{5^2}}}} + {frac{{146}}{{{5^3}}}} cr
& = 29 + 5 + 1 cr
& = 35 cr} $$ Note: We have taken integral value only, not the fractional. For example $$frac{{146}}{5}$$ = 29.2 but we have taken 29 and so on.