5. IMP RULE - Divisibility Trick: What is the remainder ?

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Shortcut rule: If exponent is even, difference of powers divisible by modulus factors → remainder = 0.

IMP RULE - Divisibility Trick

If a number is of the form (aⁿ + bⁿ):

Exam safe memory trick Divisibility Rules

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

Ex: Q. 29¹⁰¹ + 31¹⁰¹

Solⁿ

• n = 101 (ODD)


• a + b = 29 + 31 = 60

Therefore: 29¹⁰¹+31¹⁰¹ is divisible by 60.

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

Ex: 56⁹⁹- 44⁹⁹

Solⁿ

n = 99 (ODD)

• a – b = 56 - 44 = 12


• a + b = 56 + 44 = 100

Therefore: 56⁹⁹ - 44⁹⁹ is divisible by 12 and 100.

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

Ex: 17⁴⁸ + 3⁴⁸

• n = 48 (EVEN)


• a + b = 30 → Not divisible


• a – b = 4 → Not divisible

Not divisible by 30 or 4.

4.  aⁿ − bⁿ , n = even: Yes Divisible by (a - b), No Not divisible by (a + b)

Ex: 81²⁰⁰−79²⁰⁰

• n = 200 (EVEN)


• a - b = 81- 79 = 2 → Divisible


• a + b = 81 + 79 = 160  → Not divisible

So: 81²⁰⁰ - 79²⁰ is divisible by 2 but not by 160.

Exam Tricks :

• ( + ) + Odd → divisible by (a + b)


• ( - ) → divisible by (a - b)


• Even powers → sum (aⁿ + bⁿ) not safe


• Even powers → difference (aⁿ - bⁿ) → (a - b)