A lady grows cabbage in her garden that is in the shape of a square. Each cabbage takes 1 square foot of area in her garden. This year, she has increased her output by 127 cabbages when compared to last year. The shape of the area used for growing the cabbage has remained a square in both these years. How many cabbages did she produce this year?  Choices.
A) 4096	B) 4098	C) 5000	D) 5096	E) Cannot be determined
Explanatory Answer   The shape of the area used for growing Cabbage has remained a square in both the years.   Let the side of the square area used for growing Cabbage this year be X ft.   Therefore, the area of the ground used for cultivation this year = X2 sq.ft.   Let the side of the square area used for growing Cabbage last year be Y ft.   Therefore, the area of the ground used for cultivation last year = Y2 sq.ft.   As the number of cabbage grown has increased by 127, the area would have increased by 127 sq ft because each cabbage takes 1 sq ft space.   Hence, X2 - Y2 = 127   (X + Y)(X - Y) = 127.   127 is a prime number and hence it will have only two factors. i.e., 127 and 1.   Therefore, 127 can be expressed as product of 2 numbers in only way = 127 * 1   i.e., (X + Y)(X - Y) = 127* 1   So, (X + Y) should be 127 and (X - Y) should be 1.   Solving the two equations we get X = 64 and Y = 63.   Therefore, number of cabbage produced this year = X2 = 642 = 4096.   Alternative Approach : Use answer choices   The area in both the years are squares of two numbers.   That rules out choice B,C and D. as 4098,5000 and 5096 are not the square of any number.   Check Choice A: If this year's produce is 4096, last year's produce would have been  4096 - 127= 
<gadget id="calculator"> 4096 - 127</gadget>
<output>3_969</output>
3969   3969 is the square of 63.   So, 4096 is the answer.   Choice 4096
<result>A</result>
A lady grows cabbage in her garden that is in the shape of a square. Each cabbage takes 1 square foot of area in her garden. This year, she has increased her output by 127 cabbages when compared to last year. The shape of the area used for growing the cabbage has remained a square in both these years. How many cabbages did she produce this year?