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Official Quant Thread for CAT 2012 [Part 1] Quantitative

Hi puys,
CAT 2011 season is finally over [smiley] [smiley]
Let start a new season of Quants thread for CAT 2012.. [smiley] [smiley]
Here are few links of Quants thread for CAT 2011 season...
Part-8
http://www.pagalguy.com/forum/quanti...at-2011-a.html (Official Quant Threa...
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Page 274 of 491
Page 275 of 491
very very urgent puys

Q) how many for digit perfect squares have the property that their last two digits are same??


sol:
only square of 2 digit result in 4 digit number
a 2 digit number ab =10a+b
square it (10a+b)^2=100a^2+b^2+20ab
last 2 digits comes from (b^2+20ab)
now for odd b unit digit same not possible since 2a is even
i will denote U as unt digit
b=0 => a=4,5,6,7,8,9 -------------> 6 numbers
b=2 => 4+40a this means U(4a)=4 =>a=6 number is 62
b=4 => 16+80a this means U(8a+1)=6 not possible since U(8a+1) always odd
b=6 => 36+120a means U(2a+3)=6 not possible
b=8 => 64+160a means U(6a+6)=4 for a=3 numberis 34

total 8 numbers
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