# A theater can seat 986 people. The number of rows is 5 less than the number of seats in each row. How many rows of seats are there?

Nov 26, 2016

The required factors of 986 are $29 \mathmr{and} 34.$
The number of rows is 29.

#### Explanation:

The maths required here is to find two factors of 986 which differ by 5. It would be unreasonable for anyone to know all the factors of 986, but this technique or strategy might be of help.

A difference of only 5 means the two factors are very close to $\sqrt{986}$

Also their product must end in a 6.

Factors which give a product ending in 6 are:
$1 \times 6 = 6 \text{ "2xx3 =6" } 2 \times 8 = 16$

$4 \times 4 = 16 \text{ "4xx9 = 36" } 6 \times 6 = 36$

$\sqrt{986} = 31.4 \ldots$

One factor must be smaller and one larger than $31.4 \ldots .$

Find two numbers which differ by 5, very close to 31.4, that have a product of 986#

$29 \times 34 = 986$ are the only possibilities, they differ by 5

$28 \times 32 = 896 \text{ } \leftarrow$ too small, differ by 4
$26 \times 36 = 936 \text{ } \leftarrow$ too small, too far apart, differ by 10

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Background knowledge ... Nice to know

Composite numbers are made up of pairs of factors: one bigger and one smaller.

The factors of 36 arranged in order are:

$1 \text{ "2" "3" "4" "6" "9" "12" "18" } 36$
$\textcolor{w h i t e}{\times \times \times \bigvee . v \times x} \uparrow$
$\textcolor{w h i t e}{\times \times x \bigvee v \times \times} \sqrt{36}$

The outer factors $\left(1 \mathmr{and} 36\right)$ have the greatest sum and the greatest difference. $1 + 36 = 37 \mathmr{and} 36 - 1 = 35$

the factors closer to the middle have a smaller sum and smaller difference. For example, $9 \mathmr{and} 4 : \text{ } 9 + 4 = 13 \mathmr{and} 9 - 4 = 5$

However the factor exactly in the middle is the square root.
$1 \text{ "2" "3" "4" "6" "9" "12" "18" } 36$
$\textcolor{w h i t e}{\times \times \times \bigvee . v \times x} \uparrow$
$\textcolor{w h i t e}{\times \times x \bigvee v \times \times} \sqrt{36}$

Factors which are the square roots have the smallest sum $6 + 6 = 12$ and the smallest difference $6 - 6 = 0$

Therefore the difference between two factors of a number is an indication of whether the factors are 'outside factors' or factors close to the middle.