Square Total
What is the smallest number Anastasia could have thought of, if after performing some operations, the total is a square number?
Age
11 to 14
Challenge level
Problem
Anastasia thinks of a positive integer.
Barry doubles Anastasia's number.
Connor trebles Barry's number.
Damion multiplies Connor's number by $6$.
Eve notices that the total of the four numbers is a perfect square.
What is the smallest positive integer that Anastasia could have thought of?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematica ideas.
Student Solutions
Answer: $5$
Anastasia: $a$
Barry: $a\times2=2a$
Connor: $2a\times3=6a$
Damion: $6a\times6=36a$
Eve: $a+2a+6a+36a=45a$ is a square number.
Using the $45$ times table
| $a$ | $45a$ | Square? |
|---|---|---|
| $1$ | $45$ | No |
| $2$ | $90$ | No |
| $3$ | $135$ | No |
| $4$ | $180$ | No |
| $5$ | $225$ | Yes: $15^2 = 225$ |
This shows that the smallest possible number would be $5$.
Alternatively, for $45a$ to be a square number, each of its prime factors must be raised to an even power. Since $45 = 3^2\times 5$ as a product of prime factors, the only prime factor not raised to an even power is $5$. Therefore the smallest value of $a$ that makes this a square must be $5$.