Category Archives: STEP and MAT

STEP I-2006-1 and mental arithmetic

STEP I/2006/1 – Find the integer, n, that satisfies n^{2} < 33\,127 < (n + 1)^2 . Find also a small integer m such that (n + m)^2 - 33\,127 is a perfect square. Hence express 33,127 in the form pq, where p and q are integers greater than 1.

By considering the possible factorisations of 33,127, show that there are exactly two values of m for which (n + m)^2 - 33\,127 is a perfect square, and find the other value.

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