25¹⁹⁰-19¹⁹⁰-8¹⁹⁰+2¹⁹⁰ is divisible by :

Solution

Maths sir · 5 steps

Pair the terms so the gaps match: 25-8 = 19-2 = 17, and the factor 17 drops right out.

1. Regroup for 17. Write the expression as (25¹⁹⁰-8¹⁹⁰) - (19¹⁹⁰-2¹⁹⁰). Since aⁿ-bⁿ is always divisible by a-b, the first bracket is divisible by 25-8=17 and the second by 19-2=17. So the whole expression is divisible by 17.
2. Check for 2. 25¹⁹⁰ and 19¹⁹⁰ are odd, while 8¹⁹⁰ and 2¹⁹⁰ are even. Odd minus odd minus even plus even is even, so the expression is divisible by 2. Since 2 and 17 share no common factor, it is divisible by 2 × 17 = 34.
3. Test 7. For 14 we need 7 as well. Modulo 7, 25 ≡ 4, 19 ≡ -2, 8 ≡ 1. Since 190 is even, (-2)¹⁹⁰ = 2¹⁹⁰, so the 2¹⁹⁰ terms cancel.

   E ≡ 4¹⁹⁰ - 2¹⁹⁰ - 1 + 2¹⁹⁰ = 4¹⁹⁰ - 1 (mod 7)

4. Finish mod 7. 4³ = 64 ≡ 1 (mod 7) and 190 = 3 × 63 + 1, so 4¹⁹⁰ ≡ 4. Then E ≡ 4 - 1 = 3 (mod 7), which is not 0. So the expression is not divisible by 7, and therefore not by 14.
5. So. The expression is divisible by 34 but not by 14, which is option (d).

The idea to keep. For sums of big powers, regroup the terms so each pair has the same difference a-b, since a-b always divides aⁿ-bⁿ. Test any other factor with remainders, never by guessing.

Answer from the NTA final answer key. Solution by PrepWiser.