questioner wrote:

Your attempt to divide by ten in the explanation does not make any sense.

We have prime-factorized 690 into 690 = 2 × 3 × 5 × 23.

These are the prime factors. Any other factor will be a combination (a product) of the given prime factors. So the divisors of 690, starting from the smallest, are:

**1**,

**2**,

**3**,

**5**,

**6** *(which is 2 × 3)*,

**10** *(which is 2 × 5)*,

**15** *(which is 3 × 5)*, etc.

We divide 690, starting by the smallest factor:

690 / 1 =

**690**690 / 2 =

*345*690 / 3 =

*230*690 / 5 =

*138*690 / 6 =

*115*690 / 10 =

*69*690 / 15 =

*46*First of all note, that on the right we get the divisors (factors) of 690. Every factor of 690,

*n*, has a corresponding factor of 690, (690/

*n*). If we continue to increase the factor we divide by we will decrease the factor on the right side of the equality. So we do not need to consider any more factors of 690 and we already have a full list of 3-digit factors of 690:

690, 345, 230, 138, 115.