Number System - Study Mode

[#491] On multiplying a number by 7 all the digit in the product appear as 3's , the smallest such numbers is -
Correct Answer

(C) 47619

Explanation

Solution: $$eqalign{
& { ext{According to question,}} cr
& x imes 7 = 333333 cr} $$ (In answer 5 digit number are given, so we take 6 digit 333333) $$eqalign{
& x = frac{{333333}}{7} cr
& x = 47619 cr} $$

[#492] Find the largest number, which exactly divides every number of the form (n 3 - n) (n - 2) where n is a natural number greater than 2.
Correct Answer

(C) 24

Explanation

Solution: $$eqalign{
& Rightarrow left( {{n^3} - n}
ight)left( {n - 2}
ight){ ext{put n = 3}} cr
& Rightarrow left( {{3^3} - 3}
ight)left( {3 - 2}
ight) cr
& Rightarrow left( {27 - 3}
ight) imes 1 cr
& Rightarrow 24 cr
& { ext{It is divisible by }}24 cr} $$

[#493] Two number when divided by 17, leaves remainder 13 and 11 respectively. If the sum of those two numbers is divided by 17, the remainder will be ?
Correct Answer

(C) 7

Explanation

Solution: Dividend = divisor × quotient + remainder $$eqalign{
& { ext{First number }} cr
& Rightarrow { ext{ }}left( {17 imes { ext{n}}}
ight) + 13 cr
& { ext{Let n = 1}} cr
& Rightarrow (17 imes 1) + 13 cr
& Rightarrow 30 cr
& { ext{second number }} cr
& Rightarrow { ext{ }}left( {17 imes { ext{n}}}
ight) + 11 cr
& Rightarrow (17 imes 1) + 11 cr
& Rightarrow 28 cr
& { ext{According to question}} cr
& = frac{{30 + 28}}{{17}} cr
& = frac{{58}}{{17}} cr
& Rightarrow { ext{remainder = 7}} cr} $$ Alternate : Divisor = 1 st remainder + 2 nd remainder - 3 rd remainder $$eqalign{
& 17 = { ext{ }}13 + 11 - { ext{3rd remainder}} cr
& { ext{3rd remainder}} = 24 - 17 = 7 cr} $$

[#494] Unit digit in $${left( {264}
ight)^{102}} + {left( {264}
ight)^{103}}$$ xa0xa0 is :
Correct Answer

(A) 0

Explanation

Solution: $${left( {264}
ight)^{102}} + {left( {264}
ight)^{103}}$$ unit digit $$eqalign{
& {{ ext{4}}^1} o 4 o 4 cr
& {4^2} o 16 o 6 cr
& {4^3} o 64 o 4 cr} $$ Rule: When 4 has odd power, then unit digit is 4 When 4 has even power, then unit digit is 6 $$eqalign{
& {left( {264}
ight)^{102}} + {left( {264}
ight)^{103}} cr
& ,,,,,, downarrow ,,,,,,,,,,,,,,,,,,,,,,,,,,,, downarrow cr
& ,,,,{4^{102}},,,,,,, + ,,,,,,{4^{103}} cr} $$ $${ ext{unit digit}} = 6 + 4 = 10 o 0$$ (even power) xa0 (odd power) $$eqalign{
& {x08f{Alternate}} cr
& Rightarrow {left( {264}
ight)^{102}} + {left( {264}
ight)^{103}} cr
& Rightarrow {left( {264}
ight)^{102}} + left( {1 + 264}
ight) cr
& Rightarrow {left( {264}
ight)^{102}} + 265 cr
& { ext{Multiple of }}5,,{ ext{and }}2 cr
& { ext{So}},{ ext{unit digit is }}0 cr} $$

[#495] How many number between 1000 and 5000 are exactly divisible by 225 ?
Correct Answer

(B) 18

Explanation

Solution: $$eqalign{
& { ext{First number = 1125}} cr
& { ext{Last number = 4950}} cr
& { ext{Number of term }} cr
& { ext{ = }}frac{{4950 - 1125}}{{225}} + 1 cr
& = frac{{3825}}{{225}} + 1 cr
& = 17 + 1 cr
& = 18 cr} $$