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King's Maths School

463: One Million and One

Can you find the right algebraic approach to factorise these large numbers?

(1) Factorise x6 + 1 as (x2 + 1)(x4 + ax3 + bx2 + cx + d), where a, b, c, d are constants to be determined.

(We might notice this is possible because (-1)3 = -1, and this means that if x2 + 1 =0 we get x6 + 1 = 0.)

(2) Write 1000001 as a product of two integer factors without using a calculator.

(3) Write 10000000001 as a product of two integer factors without using a calculator.

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