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Questions
 A piece of land is to be divided into 20 acres or 24 acres or 26 acres or 28 acres for farming and leave 7 acres for grazing. Determine the smallest size of such land.
 When a certain number x is divided by 30, 45 or 54, there is always a remainder of 21. Find the least value of the number x
 A number m is such that when it is divided by 30, 36, and 45, the remainder is always 7. Find the smallest possible value of m.
 Find the L.C.M of x^{2} + x , x^{2} – 1 and x^{2} – x
Answers

2  20  24  26  28 
2  10  12  13  14 
2  5  6  13  7 
3  5  3  13  7 
5  5  1  13  7 
7  1  1  13  7 
13  1  1  13  1 
1  1  1  1  1 
Size of the land = ((2^{3} x 3 x 5 x 7 x 13) + 7) acres
= 10920 + 7 = 10,927 acres 2  30  45  54 
3  15  45  27 
3  5  15  9 
3  5  5  1 
5  5  5  1 
1  1  1  1 
Least value of x = (2 x 3^{3} x 5) + 21
=270 + 21 = 291 2  30  36  45 
2  15  18  45 
3  15  9  45 
3  5  3  15 
5  5  1  5 
 1  1  1 
Smallest value of m = (2^{2} x 3^{3} x 5) + 7 = 187 x^{2} + x = x (x + 1)
x^{2} 1 = (x + 1) (x – 1)
x^{2} – x = x(x – 1)
x(x+1) (x – 1)
x^{3} x