Question

Question #24c58

Answer 1

`41 xx 41 =1681 and 41+41=82`

Explanation:

Factors of a number are always in pairs.

The further apart the factors are, the greater is their sum and their difference. The closer they are, the less is their sum and their difference.

Consider `36`:
The factor pairs and the sum and difference are as follows:

`"Factors" color(white)(...........)"sum"color(white)(...........)"difference"`

`1xx36color(white)(...............)37color(white)(.................)35`
`2xx18color(white)(...............)20color(white)(.................)16`
`3xx12color(white)(...............)15color(white)(......l..xxx..)9`
`4xx9color(white)(................)13color(white)(......l..xxx..)5`
`6xx6color(white)(................)12color(white)(......l..xxx..)0`

Notice that as the factors get closer to the square root, the sum and the difference decrease.

`sqrt1681 =41" rArr41 xx 41 = 1681`

`41+41 = 82`

This will be the smallest sum of any factors.

Note that the largest sum `= 1681 +1 =1682`