3. Divisibility of Algebraic Expressions: a^n ± b^n

These identities are extremely common in “which of the following divides…” type SSC/RRB questions.

Rule 1: a^n - b^n is always divisible by (a-b), for every positive integer n. Rule 2: a^n - b^n is divisible by (a+b) only when n is even. Rule 3: a^n + b^n is divisible by (a+b) only when n is odd. Rule 4: a^n + b^n is never divisible by (a-b) in general (no standard guarantee) — don’t apply a rule here.

Reasoning (why Rule 1 works): a^n - b^n = (a-b)(a^(n-1)+a(n-2)b+…+b^(n-1)), a standard factorization, so (a-b) is always a factor.

Reasoning (Rule 2): When n is even, write n=2m: a^(2m)-b(2m) = (a^m-bm)(a^m+bm). Since this can be regrouped so that (a+b) appears as a factor of a^n - b^n when n is even (verify via the general factorization a^n-bn=(a+b)(a^(n-1)-a(n-2)b+a^(n-3)b2-…-b^(n-1)) when n is even, the terms telescope to divide out evenly).

Reasoning (Rule 3): For odd n, a^n+bn = (a+b)(a^(n-1)-a(n-2)b+a^(n-3)b2-…+b^(n-1)) — a valid factorization only when n is odd.

Example 1: Is 7^6 - 3^6 divisible by 7+3=10? Since n=6 is even, by Rule 2, yes. (Verified: 7^6-36=117649-729=116920, and 116920/10=11692 exactly.)

Example 2: Is 5⁷⁺²7 divisible by 5+2=7? Since n=7 is odd, by Rule 3, yes. (Verified: 5⁷⁺²7=78125+128=78253=7×11179.)

Example 3: Is x^(15)-y(15) divisible by x-y and by x+y? By Rule 1, always divisible by (x-y). Since n=15 is odd, NOT guaranteed divisible by (x+y) (Rule 2 needs even n).

Exam application: A very common SSC question: “5⁽⁹⁹⁾⁺⁷(99) is divisible by which of the following?” → since exponent 99 is odd, it’s divisible by 5+7=12 (and any factor of 12: 2,3,4,6).

Quick-reference table:

Example 4: Is 11^(20)-9(20) divisible by both 2 and 20? Here a-b=11-9=2 and a+b=11+9=20, and n=20 is even. By the table above, both (a-b)=2 and (a+b)=20 divide a^n-bn when n is even. So yes, 11^(20)-9(20) is divisible by both 2 and 20.

Example 5 (odd exponent, sum form): Is 9⁷⁺⁴7 divisible by 13? Here a+b=9+4=13, and n=7 is odd, so by Rule 3, 9⁷⁺⁴7 is divisible by 13. Verification: 9^7=4,782,969 and 4^7=16,384, so 9⁷⁺⁴7=4,799,353, and 4,799,353 ÷ 13 = 369,181 exactly. ✓

Example 6 (larger co-prime pair, odd exponent): Is 13⁽¹¹⁾⁺⁷(11) divisible by 20? Here a+b=13+7=20, and n=11 is odd, so by Rule 3 it must be divisible by 20 — no need to compute the actual (enormous) value of 13⁽¹¹⁾⁺⁷(11) at all; this is exactly the time-saving power of the rule. (A computational check confirms 13⁽¹¹⁾⁺⁷(11) ≡ 0 (mod 20).)

Example 7 (harder — even exponent, difference form, two divisors at once): Is 12^(10)-5(10) divisible by both 7 and 17? Here a-b=12-5=7 and a+b=12+5=17. Since n=10 is even, by the table both (a-b)=7 and (a+b)=17 divide a^(10)-b(10). So 12^(10)-5(10) is divisible by both 7 and 17 — and consequently also by 7×17=119, since 7 and 17 are co-prime. (Computational check confirms both remainders are 0.)