Number System - Study Mode

[#461] The remainder obtained when any prime number greater than 6 is divided by 6 must be :
Correct Answer

(C) either 1 or 5

Explanation

Solution: Let the required prime number be p. Let p when divided by 6 give n as quotient and r as remainder. Then p = 6n + r, where 0 $$ leqslant $$ r < 6 Now, r = 0, r = 2, r = 3 and r = 4 do not give p as prime. ∴ r $$
e $$ 0, r $$
e $$ 2, r $$
e $$ 3, and r $$
e $$ 4 Hence, r = 1 or r = 5

[#462] The smallest number which must be subtracted from 8112 to make it exactly divisible by 99 is :
Correct Answer

(C) 93

Explanation

Solution: On dividing 8112 by 99, we get 93 as remainder. So, the required number to be subtracted is 93.

[#463] If the sum of two numbers is 14 and their difference is 10, find the product of these two numbers.
Correct Answer

(C) 24

Explanation

Solution: Let the two numbers be are a and b ∴ a + b = 14.....(i) a - b = 10.....(ii) By adding equation (i) and (ii) we get 2a = 24 ∴ a = 12 and b = 2 ∴ Product of these two numbers = 12 × 2 = 24

[#464] The greatest number by which the product of three consecutive multiples of 3 is always divisible is :
Correct Answer

(C) 162

Explanation

Solution: Three consecutive multiples of 3 are 3m, 3(m + 1) and 3(m + 2) Their product = 3m × 3(m + 1) × 3(m + 2) = 27 × m × (m + 1) × (m + 2) Putting m = 1, this product is (27 × 1 × 2 × 3) = 162 So, this product is always divisible by 162

[#465] If p 3 - q 3 = (p - q) (p - q) 2 - xpq, then find the value of x :
Correct Answer

(B) - 3

Explanation

Solution: $${p^3} - {q^3} = left( {p - q}
ight)$$ xa0 $$left{ {{{left( {p - q}
ight)}^2} - xpq}
ight}$$ $$ Rightarrow left( {p - q}
ight)left( {{p^2} + {q^2} + pq}
ight)$$ xa0 xa0 $$ = left( {p - q}
ight)$$ xa0 $$left{ {{{left( {p - q}
ight)}^2} - xpq}
ight}$$ $$left{ {x08ecause {a^3} - {b^3} = left( {a - b}
ight)left( {{a^2} + ab + {b^2}}
ight)}
ight}$$ By cancelling same terms of both sides $$ Rightarrow {p^2} + {q^2} + pq = {p^2} + {q^2}$$ xa0 xa0 $$ - 2pq - xpq$$ xa0 $$left{ {{{left( {a - b}
ight)}^2} = {a^2} + {b^2} - 2ab}
ight}$$ $$eqalign{
& Rightarrow 3pq = - xpq cr
& Rightarrow x = - 3 cr} $$