Question

The number among the following that divides 7 ¹² - 2²⁴
1) 32
2)33
3) 34
4)35

Answer 1

7^12 - 2^24 = (7^6)^2 - (2^12)^2                 [$a^{2} - b^{2} = (a+b)(a-b)$]

= (7^6 + 2^12){(7^3)^2 - (2^6)^2}

= (7^6 + 2^12){(7)^3 + (2^2)^3}{(7)^3 - (2^2)^3}

[$a^{3} + b^{3} = (a+b)(a^{2} + b^{2} - ab)$] [$a^{3} - b^{3} = (a-b)(a^{2} + b^{2} + ab)$]

= (7^6 + 2^12)(7 + 4)(49 + 4 - 28)(7 - 4)(49 + 4 + 28)

[$a^{2} - b^{2} = (a+b)(a-b)$]

= (7^6 + 2^12)(49 + 4 - 28)(49 + 4 + 28)(49 - 16)

Since, 49 - 16 = 33 and it is one of the options.

∴ Ans: 2) 33

Hope I helped

PS, I have given the formulas to make it easier to understand. In these types of questions, you should simplify by formulas until you find a suitable pair matching the answer. Cz it is a 1 mark Q(should be), you can write the ans only