Xi'an's Og

Each block in the pyramid above is the product of the two blocks below it, a, b, c, and d are distinct integers that are not perfect squares. What values for a, b, c, and d result in a integer top square value? What’s the smallest possible integer value for the top square?

Multiplying terms to the top of the pyramid return bc√(abcd), hence abcd must be a square x² with none of a,b,c,d being a square. If x=p¹p²p³p⁴ (with all terms distinct but not necessarily prime) and a=p¹p², b=p²p³,c=p³p⁴, and d=p⁴p¹, this works out, unless one of the terms a,b,c,d is a square. For instance, x=24=4x2x1x3 works, with a=8,b=2,c=3 and d=12, resulting in a top value of 6×24=144. Actually, the same applies to x=p¹p²p³ and a=p¹p², b=p¹,c=p³, and d=p²p³, with p²>max(p¹,p³). Then x=2x3x4=24 leads to a pyramid result of 2x3x24=144, which could be the smallest possible value.

This entry was posted on May 13, 2024 at 12:24 am and is filed under Books, Kids with tags arithmetics, mathematical puzzle, riddle, The Fiddler. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.