Number System - Study Mode

[#306] $$999,frac{1}{7} + 999frac{2}{7} + 999frac{3}{7} + $$ xa0 xa0 $$999frac{4}{7} + $$ xa0 $$999frac{5}{7} + $$ xa0 $$999frac{6}{7}$$ xa0 is simplified to-
Correct Answer

(A) 5997

Explanation

Solution: $$999,frac{1}{7} + 999frac{2}{7} + 999frac{3}{7} + $$ xa0 xa0 $$999frac{4}{7} + $$ xa0 $$999frac{5}{7} + $$ xa0 $$999frac{6}{7}$$ $$ Rightarrow left( {999 imes 6}
ight) + $$ xa0 $$left( {frac{1}{7} + frac{2}{7} + frac{3}{7} + frac{4}{7} + frac{5}{7} + frac{6}{7}}
ight)$$ $$eqalign{
& Rightarrow 5994 + left( {frac{{21}}{7}}
ight) cr
& Rightarrow 5994 + 3 cr
& Rightarrow 5997 cr} $$

[#307] The unit digit in $${left( {122}
ight)^{173}}$$xa0 is ?
Correct Answer

(A) 2

Explanation

Solution: In this case: 122 (powers of the last digit 2 ) 2 1 = 2 2 2 = 4 2 3 = 8 2 4 = 16 2 5 = 32 2 6 = 64 2 7 = 128 2 8 = 256 2 9 = 512 …… As you can see, the pattern repeats every 4 th power (2, 4, 8, 6). We have to find 122 173 Hence, remainder when 173 is divided by 4 = 1 (which corresponds to the 1 st number of the pattern) Thus, unit digit of 122 173 = 2

[#308] $$,{2^{16}} - 1$$ xa0 is divisible by -
Correct Answer

(C) 17

Explanation

Solution: $$eqalign{
& Rightarrow {2^{16}} - 1,, cr
& = left( {{2^8} - {1^8},,}
ight)left( {{2^8} + {1^8},}
ight) cr
& = left( {{2^4} - 1,,}
ight)left( {{2^4} + 1,,}
ight)left( {{2^8} + 1,,}
ight) cr
& = left( {16 - 1}
ight)left( {16 + 1}
ight)left( {{2^8} + 1,,}
ight) cr
& = 15 imes 17left( {{2^8} + 1,,}
ight) cr
& herefore {2^{16}} - 1,,{ ext{is divisible by 17}} cr} $$

[#309] If a = 4011 and b = 3989, then value of ab = ?
Correct Answer

(A) 15999879

Explanation

Solution: a = 4011 b = 3989 a × b = 4011 × 3989 = 15999879 Shortcut method : a = 4011 ⇒ (4000 + 11) b = 3989 ⇒ (4000 - 11) [(a - b) (a + b) = a 2 - b 2 ] According to the question, a × b = (4000 + 11) (4000 - 11) = 4000 2 -11 2 = 16000000 - 121 = 15999879

[#310] 5349 is added to 3957. Then 7062 is subtracted from the sum. The result is not divisible by :
Correct Answer

(C) 7

Explanation

Solution: 5349 + 3957 - 7062 = 2244 This is not divisible by 7.