Expanding Zeros
The following sequence continues indefinitely... Which of these integers is a multiple of 81?
The following sequence continues indefinitely:
27 = 3 x 3 x 3
207 = 3 x 3 x 23
2 007 = 3 x 3 x 223
20 007 = 3 x 3 x 2223.......
Which of the following integers is a multiple of 81?
a) 20 000 007
b) 2 000 000 007
c) 200 000 000 007
d) 20 000 000 000 007
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Answer: d) 20 000 000 000 007
$$\underbrace{3\times3}_{=\ 9}\times\underbrace{22...23}_{\begin{array}{l}\text{must be a}\\
\text{multiple of 9}\end{array}}$$
A number is a multiple of $9$ when its digits add up to a multiple of $9$
$2+2+3=7$
$2+2+2+3=9$ so $2223$ is a multiple of $9$
Adding more $2$s, it will always be odd
$\therefore$ next possible multiple of $9$ will be $27$ with one $3$ and twelve $2$s
Number of $2$s in $22...23$ equals the number of $0$s in $20...07$
$\therefore$ need three $0$s or twelve $0$s (or twenty-one, etc)
a) six $0$s
b) eight $0$s
c) ten $0$s
d) twelve $0$s