model 2 divisibility, multiples, add and subtract based number system Section-Wise Topic Notes With Detailed Explanation And Example Questions

Answer: (d)

The expression 2⁶ⁿ − 4²ⁿ = (2⁶)ⁿ − (4²)ⁿ

= 64ⁿ − 16ⁿ

Which is divisible by 64 –16= 48

Therefore the required answer is 48.

Answer: (b)

Let the number be p.

According to the question,

∴ $\text"p + 12"/6 = 112$

⇒   p + 12 = 672

⇒   p = 672 – 12 = 660

∴  Correct answer = $\text"600"/6 = 112$

Correct answer =  110 + 12 = 122

Answer: (c)

The number (x y z x y z) can be written, after giving corresponding weightage of the places at which the digits occur, as 100000 x + 10000y + 1000z + 100x + 10y + z

= 100100x + 10010y + 1001z

= 1001 (100x + 10y + z)

Since 1001 is a factor, the number is divisible by 1001.

7 × 11 × 13 = 1001

As the number is divisible by 1001, it will also be divisible by all three namely, 7, 11 and 13 and not by only one of these because all three are factors of 1001.

So, the answer is 1001.

Answer: (b)

Here, the first divisor (91) is a multiple of the second divisor (13).

∴ Required remainder = Remainder obtained on dividing 17 by 13

⇒ 17 = ( 13 × 1 ) + 4

Hence Required remainder = 4

Answer: (c)

Answer: (c)

$a^4 - b^4 = (a - b) (a + b) (a^2 + b^2)$,

Where a and b are odd positive integers.

If two positive integers are odd, then their sum, difference and sum of their squares are always even.

∴ (a - b) (a + b) and $(a^2 + b^2)$ are divisible by 2.

Hence (a - b) (a + b) x $(a^2 + b^2) = a^4 - b^4$ is always divisible by $2^3 = 8$