18. Which of these statements is true?
A $15614=1+5^{6}-1 \times 4$
B $15615=1+5^{6}-1 \times 5$
C $15616=1+5^{6}-1 \times 6$
D $15617=1+5^{6}-1 \times 7$
E $15618=1+5^{6}-1 \times 8$
Solution
E In the context of the JMC we can assume that just one of the given options is correct, so we can find which it is by eliminating the ones that are wrong. We can do this by just considering the last digit (the units digit) of the given options.
The last digit of $5^{6}$ is 5 . Since $1+5^{6}-1 \times 4=1+5^{6}-4$ its last digit is the same as that of $1+5-4$, that is, 2 . So option A is not the correct answer. In a similar way, it follows that the last digit of $1+5^{6}-1 \times 5$ is 1 , the last digit of $1+5^{6}-1 \times 6$ is 0 , and the last digit of $1+5^{6}-1 \times 7$ is 9 . So options B, C and $\mathrm{D}$ are also not correct This leaves $\mathrm{E}$ as the only possible correct option.
In this way there is no need to evaluate $5^{6}$. However, to give a complete answer we would need to check that $\mathrm{E}$ is correct. This is straightforward:
$$
1+5^{6}-1 \times 8=1+15625-8=15626-8=15618
$$