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# Square Total

**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**

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Age 11 to 14

ShortChallenge Level

- Problem
- Solutions

Anastasia: $a$

Barry: $a\times2=2a$

Connor: $2a\times3=6a$

Damion: $6a\times6=36a$

Eve: $a+2a+6a+36a=45a$ is a square number.

$a$ | $45a$ | 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 $5$.

This problem is taken from the UKMT Mathematical Challenges.

You can find more short problems, arranged by curriculum topic, in our short problems collection.