Find a pair of factors for each number by using the difference of two squares.

a. 45
b. 77
c. 112

To find a pair of factors for each number using the difference of two squares, we need to express each number as the difference of two perfect squares.

a. Number: 45
To express 45 as the difference of two squares, we can start by finding the square root of 45, which is approximately 6.708. Since the square root is not a perfect integer, we can try finding the two closest perfect squares on either side of 45, which are 6^2 = 36 and 7^2 = 49. Taking the difference between these two squares, we have 49 - 36 = 13. Therefore, we can express 45 as the difference of two squares: 49 - 36 = 45.

The pair of factors for 45 using the difference of two squares would be (7 + 6)(7 - 6), which simplifies to (13)(1).

b. Number: 77
To express 77 as the difference of two squares, we start by finding the square root of 77, which is approximately 8.775. The two closest perfect squares on either side of 77 are 8^2 = 64 and 9^2 = 81. The difference between these two squares is 81 - 64 = 17. Therefore, we can express 77 as the difference of two squares: 81 - 4 = 77.

The pair of factors for 77 using the difference of two squares would be (9 + 4)(9 - 4), which simplifies to (13)(5).

c. Number: 112
To express 112 as the difference of two squares, we find the square root of 112, which is approximately 10.583. The two closest perfect squares on either side of 112 are 10^2 = 100 and 11^2 = 121. The difference between these two squares is 121 - 100 = 21. Therefore, we can express 112 as the difference of two squares: 121 - 9 = 112.

The pair of factors for 112 using the difference of two squares would be (11 + 3)(11 - 3), which simplifies to (14)(8).

hints:

2²-1²=4-1=3
3²-2²=9-4=5
4²-3²=16-9=7
5²-4²=25-16=9
...
So difference of squares of adjacent integers are odd numbers increasing by 2.

Similarly, difference of adjacent odd or even numbers are multiples of 4 and increase by 4's.
2²-0²=4
3²-1²=8
4²-2²=12
5²-3²=16
6²-4²=20
...