A lady grows Cauliflower in her garden that is in the shape of a square. Each Cauliflower takes 1 square foot of area in her garden. This year, she has increased her output by 223 Cauliflower when compared to last year. The shape of the area used for growing the Cauliflower has remained a square in both these years. How many Cauliflowers did she produce this year?
Choose the correct choice from the following.
A) 12541 B) 12542 C) 12544 D) 12545 E) Cannot be determined
Explanatory Answer   The shape of the area used for growing Cauliflower has remained a square in both the years.   Let the side of the square area used for growing Cauliflower 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 Cauliflower last year be Y ft.   Therefore, the area of the ground used for cultivation last year = Y2 sq.ft.   As the number of Cauliflower grown has increased by 223, the area would have increased by 223 sq ft because each Cauliflower takes 1 sq ft space.   Hence, X2 - Y2 = 223   (X + Y)(X - Y) = 223.   223 is a prime number and hence it will have only two factors. i.e., 223 and 1.   Therefore, 223 can be expressed as product of 2 numbers in only way = 223* 1   i.e., (X + Y)(X - Y) = 223 * 1   So, (X + Y) should be 223 and (X - Y) should be 1.   Solving the two equations we get X = 112 and Y = 111.   Therefore, number of Cauliflower produced this year = X2 = 1122 = 12544.   Alternative Approach : Use answer choices   The area in both the years are squares of two numbers.   That rules out choice A,B and D. as 12541,12542 and 12545 are not the square of any number.   Check Choice C: If this year's produce is 12544, last year's produce would have been  12544 - 223= 
<gadget id="calculator"> 12544 - 223</gadget>
<output>12_321</output>
12321   12321 is the square of 111.   So, 12544 is the answer.   12544hoice 12544.
<result>C</result>
A lady grows Cauliflower in her garden that is in the shape of a square. Each Cauliflower takes 1 square foot of area in her garden. This year, she has increased her output by 223 Cauliflower when compared to last year. The shape of the area used for growing the Cauliflower has remained a square in both these years. How many Cauliflowers did she produce this year?