18. The number 2024 may be split into its first two digits and its last two digits to form the numbers 20 and 24. The highest common factor of these numbers, $\operatorname{HCF}(20,24)$ is equal to 4. Similarly, 2025 may be split, and $\operatorname{HCF}(20,25)=5$.
For how many remaining years this century (i.e. after 2025 and up to and including 2099) will this highest common factor be equal to one?
Solution
030

Since $20=2 \times 2 \times 5$ any year with last two digits ' $\mathrm{YY}$ ' that form neither an even number nor a multiple of five will have $H C F(20, Y Y)=1$.
That is, any year ending in $1,3,7$, or 9 .
This decade there are two such years remaining (namely 2027 and 2029), while the remaining seven decades of this century have four such years each.
Therefore, there are $2+7 \times 4=30$ such years.