Puzzles

Alex lives in Maidstone, Kent (1) and wants French hens (3).

Ben lives in Leighton Buzzard, Beds (8) and wants gold rings (5).

Carol lives in Burcot, Worcestershire (2) and wants calling birds (4).

If \(p\) and \(q\) are prime numbers, then the number \(p^a\times q^b\) will have \((a+1)(b+1)\) factors. This is because all its factors are of the form \(p^\alpha\times q^\beta\), with \(\alpha=0,1,...,a\) and \(\beta=0,1,...,b\). The same idea can be used on numbers with three or more prime factors; in general the number \(p_1^{a_1}\times...\times p_n^{a_n}\) has \((a_1+1)\times...\times(a_n+1)\) factors.

28 can be written as: 28, 14×2, 7×4, or 7×2×2. Therefore the following numbers have 28 factors:

and any other number with 28 factors will have larger prime factors, so will be larger.

These numbers are 134217728, 24576, 1728 and 960. Therefore the smallest number with 28 factors is 960.

120

294

496

You can find my write up of a solution to this here. Pedro Freitas (@pj_freitas) sent me the following nice alternative solution:

Any number can be written as \(10a+27b\) with integer \(a\) and \(b\), since \(1=3\times27-8\times10\). So the problem may be thought of as asking when one of \(a\) and \(b\) must be negative.

Given one way of writing a number, you can get the others by shifting by 14*29. For example,

So the question now becomes: When does this adjustment fail to eliminate negative numbers?

This is when you are at what Pedro calls "limit coefficients":

So the answer is 233.

Let \(n\) and \(m\) be integers. What is the largest number that cannot be written in the form \(na+mb\), where \(a\) and \(b\) are nonnegative integers?

268

154