Multiplication Table Puzzle
In the multiplication table on the right, only some of the numbers have been given. What is the value of A+B+C+D+E?
Age
11 to 14
Challenge level
Problem
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In the multiplication table on the right, the row and column headings are all missing, and only some of the products in the table are filled in.
All the numbers in the table are positive integers.
What is the value of $A+B+C+D+E$?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Answer: 161
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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 $5$
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$8\times5=40$ and $6\times3=18$
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Using the numbers in the grid then allows us to work out more of the input factors in the table
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Then, this allows the rest of the table to be filled in, including the values of $A$, $B$, $C$, $D$ and $E$.
Therefore, $A+B+C+D+E = 6+25+48+40+42 = 161$.
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The rest of the numbers can also be filled in:
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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?