When a certain perfect square is increased by 148, the result is another perfect square. What is the value of the original perfect square? Answers:
A) 1296  B) 1369  C) 1681  D) 1764  E) 2500

Rationale: let ’ s call the two perfect squares x ^ 2 and y ^ 2 , respectively . then the given information translates as x ^ 2 + 148 = y ^ 2 . subtracting x ^ 2 gives 148 = y ^ 2 − x ^ 2 , a difference of squares . this , in turn , factors as ( y + x ) ( y − x ) = 148 . the next step is tricky . it begins with factoring 148 , which breaks down as 2 ∗ 2 ∗ 37 . since we ’ re dealing with perfect squares , x and y are positive integers , and ( y + x ) and ( y − x ) must be paired integer factors of 148 . the options are 148 ∗ 1,74 ∗ 2 , and 37 ∗ 4 . but our number properties establish that ( y + x ) and ( y − x ) must be either both odd or both even , so only 74 ∗ 2 is an actual possibility . and because for any positive integers ( y + x ) > ( y − x ) , we can conclude that y + x = 74 and y – x = 2 . solving by elimination , 2 y = 76 , y = 38 , and x = 36 . finally , we just need to square 36 . but rather than multiplying it out , note that 36 ^ 2 ends in 6 – as does only one answer , choice a . this answer must be the one we want .
Answer: A