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This investigation looks at finding the largest product when two or more positive integers have a given sum.
For the positive integers 2 and 5
• the sum $2 + 5$ is 7
• the product $2 \times 5$ is 10.
(a) Complete this table for all the different pairs of positive integers that have a sum of 8.
[Table_1]
Write down the calculation that gives the largest product.
(b) Complete this table for all the different pairs of positive integers that have a sum of 10.
Note that 3 and 7 is the same pair as 7 and 3.
[Table_2]
Write down the calculation that gives the largest product.
(c) Find the largest product of two positive integers that have a sum of 6.
(d) Use your answers to part (a), part (b) and part (c) to help you complete the table.
[Table_3]
(e) (i) The sum of two positive integers is $S$.
$S$ is an even number.
Find an expression, in terms of $S$, for the largest product of the two integers.
(ii) The sum of two positive integers is 62.
Find the largest product of the two integers.
(f) The sum of two positive integers is $S$.
$S$ is an even number.
The largest product of the two integers is 576.
Find the value of $S$.
506
539
576
(a) Complete this table for all the different pairs of positive integers that have a sum of 9. Note that 2 and 7 is the same pair as 7 and 2.
[Table_1]
Write down the calculation that gives the largest product.
(b) Find the largest product of two positive integers that have a sum of 7.
(c) Use your answers to part (a) and part (b) to help you complete the table.
[Table_2]
(d) The sum of two positive integers is $S$. $S$ is an odd number.
(i) Explain why the largest product of the two integers is always even.
(ii) Find an expression, in terms of $S$, for the largest product of the two integers. Do not simplify your answer.
548
592
523
(a) Three positive integers have a sum of 6.
Complete the table for all the different sets of positive integers that have a sum of 6. Writing the positive integers in a different order does not give a different set.
[Table_1]
Integers | Sum | Product
-----------------
| 6 | |
| 6 | |
| 6 | |
Write down the calculation that gives the largest product.
(b) Look at how you found the largest product in part (a) and in question 1(a).
Four positive integers have a sum of 40.
Show that the largest product of these four integers is 10 000.
(c) Complete the table.
[Table_2]
Sum | 6 | 40 | 15 | 24
--------------------------------
Number of positive integers in the sum | 3 | 4 | 5 | 6
--------------------------------
Largest product | | 10 000 | |
(d) $n$ integers have a sum of 40, where $n$ is a factor of 40.
Find the value of the largest product of the $n$ integers.
547
529
524
