Which of the following number always divides the difference between the square of two consecutive odd numbers?

Let smallest number = x.
Then, x + x + 1 + x + 2 + x + 3 + x + 4 = 170
? 5x + 10 = 170
? x = 160 / 5
? x = 32

? product of smallest and largest number = 32 x 36
= 1152

Let 5 consecutive even number be A = x, B = x+2, C = x+4, D = x + 6 and E = x + 8.

According to question
x + x +2 + x + 4 + x + 6 + x + 8 = 130
? 5x + 20 = 130
? x = 22
? A = 22, E = 22 + 8 = 30.
? Multiplication of A and E = 22 x 30
= 660

There can be infinite number of rational numbers between 1 and 1000.

Let be the ten's digit be x and unit's digit be y.
So the two digit number = 10x + y ( where x > y )
According to question
x + y = 14 ....(i)
x - y = 2 ....(ii)

Solving Eqs. (i) and (ii), we get
x = 8 and y = 6
? Required product = 8 x 6 = 48

9856 x 156 + 9856 x 844 = 9856 x (156 + 844)
= 9856 x 1000
= 9856000

6674 x 625 = 6674 x (5)⁴
= 6674 x (10⁴/2⁴
= 66740000/16 = 4171250

Exp.= 9998 x 999
= 9998 x (1000-1)
= 9998 x 1000 - 9998 x 1
= 9998000 - 9998 = 9988002

Let, 8888 +7777 + 666 + x = 19000
Then, 17331 + x = 19000
? x = 19000 - 17331 = 1669

Let, 4003 - x - 1599 = 716
Then, x = 4003 - 1599 - 716
= 4003 - (1599 + 716)
= 4003 - 2315
= 1688