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# Multiplication Table Puzzle

**Answer**: 161

First column: the input number at the top must be a factor of both $15$ and $18$

So it must be $1$ or $3$

Second row: the input number on the left must be a factor of both $15$ and $40$

So it must be $1$ or $5$

$(1$ or $3)\times(1$ or $5)=15$, so must be $3$ and $5$

$8\times5=40$ and $6\times3=18$

Using the numbers in the grid then allows us to work out more of the input factors in the table

Then, this allows the rest of the table to be filled in, including the values of $A$, $B$, $C$, $D$ and $E$.

Therefore, $A+B+C+D+E = 6+25+48+40+42 = 161$.

The rest of the numbers can also be filled in:

An interesting follow on question is to see what happens when the numbers are not required to be integers.

The table on the right shows a different way of completing the table if integers are not required. What has changed from the original solution? What has stayed the same?

Will this be the case for all the solutions? Can you explain why?

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Age 11 to 14

ShortChallenge Level

- Problem
- Solutions

First column: the input number at the top must be a factor of both $15$ and $18$

So it must be $1$ or $3$

Second row: the input number on the left must be a factor of both $15$ and $40$

So it must be $1$ or $5$

$(1$ or $3)\times(1$ or $5)=15$, so must be $3$ and $5$

$8\times5=40$ and $6\times3=18$

Using the numbers in the grid then allows us to work out more of the input factors in the table

Then, this allows the rest of the table to be filled in, including the values of $A$, $B$, $C$, $D$ and $E$.

Therefore, $A+B+C+D+E = 6+25+48+40+42 = 161$.

The rest of the numbers can also be filled in:

An interesting follow on question is to see what happens when the numbers are not required to be integers.

The table on the right shows a different way of completing the table if integers are not required. What has changed from the original solution? What has stayed the same?

Will this be the case for all the solutions? Can you explain why?

This problem is taken from the UKMT Mathematical Challenges.

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