IMP RULE - Divisibility Trick, : aⁿ + bⁿ , n = odd or even

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IMP RULE - Divisibility Trick

If a number is of the form (aⁿ + bⁿ):

 1.  aⁿ + bⁿ , n = odd: Yes - Divisible by (a + b), No - Not divisible by (a - b)

      2.  aⁿ − bⁿ , n = odd: Yes - Divisible by (a - b), Yes - Divisible by (a + b)

      3.  aⁿ + bⁿ , n = even: No - Not divisible by (a + b), No - Generally not divisible by (a - b)

     4.  aⁿ − bⁿ , n = even: Yes - Divisible by (a - b), Yes - Not divisible by (a + b)

Exam Trick: Plus (+) with ODD → divisible by (a + b), Minus (−) → always divisible by (a − b)

Exam Shortcut Trick: 1.( + ) + Odd → divisible by (a + b), 2. ( - ) - Odd→ divisible by (a − b), 3. Even powers → sum (aⁿ + bⁿ) not safe, 4. Even powers → difference (aⁿ − bⁿ) → (a − b)

IMP RULE: Any number multiplied by the divisor will give remainder 0 when divided by that divisor.

Ex: 12 × 7 ÷ 7  remainder = 0, x × m ÷ m remainder = 0

Q1: 25³ + 8³ ÷ 33 → Find remainder

Solⁿ

1st: It is of the form a³ + b³, where

a = 25, b = 8, n = 3 (ODD)

2^nd : Apply IMP Rule

For aⁿ + bⁿ when n is odd , Divisible by (a + b)

Here, a + b = 25 + 8 = 33

Divisor = 33

3rd : 25³ + 8³ is divisible by 33,
Remainder = 0

Ans: Remainder = 0

Exam Trick:  If number is a³ + b³ and divisor = a + b, then remainder = 0

Example:

Find the Remainder odd & even number (IMP RULE)

Q. 341²¹⁸ - 156²¹⁸ ÷ 185

Solⁿ

Reduce numbers modulo 185

= (341+156) (341−156) / 185

= 497 * 185 / 185

Remainder = 0​

TRICK: If a ≡ b (mod m) ⇒ aⁿ−bⁿ ≡ 0 (mod m)

Q. 341²¹⁸ - 156²¹⁸ ÷ 185

1st: It is of the form a³ - b³, where

a = 341, b = 156, n = 218 (even)

2^nd : Apply IMP Rule

For aⁿ - bⁿ when n is even , Divisible by (a + b) (a - b)

Here, a + b = 341 + 156 = 497, a - b = 341 - 156 = 185

Divisor = 185

3rd : 497 x 185 is divisible by 185,
Remainder = 0

TRICK: If a ≡ b (mod m) ⇒ aⁿ−bⁿ ≡ 0 (mod m)