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 211 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?
Choose the correct answer:
A) 11236
B) 11025
C) 14400
D) 12696
E) Cannot be determined
Explanatory Answer   The shape of the area used for growing cabbages has remained a square in both the years.   Let the side of the square area used for growing cabbages 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 cabbages last year be Y ft.   Therefore, the area of the ground used for cultivation last year = Y2 sq.ft.   As the number of cabbages grown has increased by 211, the area would have increased by 211 sq ft because each cabbage takes 1 sq ft space.   Hence, X2 - Y2 = 211   (X + Y)(X - Y) = 211.   211 is a prime number and hence it will have only two factors. i.e., 211 and 1.   Therefore, 211 can be expressed as product of 2 numbers in only way = 211 * 1   i.e., (X + Y)(X - Y) = 211 * 1   So, (X + Y) should be 211 and (X - Y) should be 1.   Solving the two equations we get X = 106 and Y = 105.   Therefore, number of cabbages produced this year = X2 = 1062 = 11236.   Alternative Approach : Use answer choices   The area in both the years are squares of two numbers.   That rules out choice D. 12696 is not the square of any number.   Check Choice A: If this year's produce is 11236, last year's produce would have been  11236 - 211= 
<gadget id="calculator"> 11236 - 211</gadget>
<output>11_025</output>
11025   11025 is the square of 105.   So, 11236 is the answer.   Choice 11236 is the correct answer.
<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 211 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?