practice problems

 Q1. aabb is a 4 digit number and a square. find it.
n = aabb = 1100a + 11b = 11(100a + b) 
=> 100a + b = 11k for 'n' to be a squre.
a = 1 to 9
b = 0 to 9
a = 1, b = 0-9 => range = 100 to 109 , nothing here divisible by 11.
a = 2,3... will give us these numbers divisible by 11:
209
308
407
506
605
704
803
902

When we divide them by 11, we should get a square.
It happens only with 704 = 11*64
So the answer is 704*11 = 7744

Q2.

Find all positive integers n less than 1999 such that n^2  is equal the cube of the sum of n's digits.
Solution:
n^2 = s^3 => s is also a square. Let s = m^2
=> n^2 = m^6 => n = m^3
So iterate over all cubes < 1999 => 1 <= m <= 12
m = 1 => m^3 = 1 => n = 1 s = 1 n^2 = s^3 True
m = 3 => m^3 = 27 = n => s = 9, s^3 = 729 = n^2 = 27^2 True
No other number works
Answer n = 1,27


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