## Teaser 2973: Something in common

**From The Sunday Times, 15th September 2019** [link]

I have written down four different numbers. The third number is the highest common factor of the first two (i.e. it is the largest number that divides exactly into both of them). The fourth number is the lowest common multiple of the first two (i.e. it is the smallest number that both of them divide exactly into).

I can consistently replace digits by letters in my numbers so that the highest common factor is

HCFand the lowest common multiple isLCM.What are the first two numbers?

[teaser2973]

## Jim Randell 4:06 pm

on13 September 2019 Permalink |The two mystery numbers have a common divisor that is 3 digits, and also the have a common multiple that is 3 digits, so they are both 3 digit numbers themselves.

Denoting the numbers

UVWandXYZwe can use the [[`SubstitutedExpression()`

]] solver from theenigma.pylibrary to give a direct interpretation of the the puzzle.This run file executes in 989ms.

Run:[ @repl.it ]Solution:The first two numbers are 278 and 417.LikeLike

## Jim Randell 5:14 pm

on13 September 2019 Permalink |Here’s an alternative approach that uses a bit of analysis:

Each of the mystery numbers must be some small (proper) multiple of

HCF, say,A×HCFandB×HCF. AndAandBmust be co-prime.We can then use the fact that:

to give a faster alphametic approach.

The following run file executes in 173ms.

Run:[ @repl.it ]Actually it is clear that

A×B< 10, so the only options areA= 2,B= 3, giving rise to the following one-line solution:LikeLike

## GeoffR 10:09 am

on17 September 2019 Permalink |LikeLike