24 24 6
Number Theory
Level
2
Is it true that among any 24 consecutive integers, all larger than 5, there always exists a number which has at least three (not necessarily distinct) prime factors?
As an example, 12 has three prime factors: 2, 2, 3.
true
false
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
We will show, that a much stronger statement is true:
"Amongst any 8 positive consecutive integers, there is at least one, which has at least three (not necessarily distinct) prime factors."
It is easy to see, that amongst 8 consecutive integers, there is exactly one, which is divisible by 8. Since 8 (= 2×2×2) itself has 3 prime factors, therefore this number has at least 3 prime factors.
Hence, our answer should be:
TRUE