Math: an interesting thing about 30

Tutoring math, number exploration is interesting. The tutor mentions 30 as an example of consecutive factors.

It’s not new to say that 30 = 5×6. However, the factor set 5×6 is interesting, in a way.

In math, n and n+1 (for instance, 5 and 6) are always relatively prime; that is, the only number that goes into both will be 1. Therefore, the only way a number has both factors n and n+1 is if both multiply together to make it (perhaps along with other numbers, depending on its size).

Another example is 84, which is 42×2 or 6x7x2. Once again, 6 and 7 both go into 84, and you need multiply them both together to make it. You can write 6 as 2×3, but it’s still there: (2×3)x7x2 = 84.

Note that this isn’t necessarily true of other patterns. For instance, n and n+2 can both go into a number, but not both be part of its factorization. 4 and 6 both go into 12, for instance, but 4×6=24 rather than 12.

Travers, Kenneth J. et al. Using Advanced Algebra. Toronto: Doubleday Canada Limited, 1977.