400 can be factored as 24 * 52, or 22 * 22 * 52. A perfect square factor of 400 will always have an even exponent, so here is a list:
12, 22, 42, 52, 102, and 202.
They are all perfect squares
496
20 squares x 20 squares = 400 squares
501
They are: 529 and 576
1 and 400.
400 and 900 are squares of 20 and 30, respectively.
16 => 256 17 => 289 18 => 324 19 => 361 20 => 400
The perfect squares between 101 and 400 are the squares of the integers from 11 to 20. These whole numbers are 121 (11²), 144 (12²), 169 (13²), 196 (14²), 225 (15²), 256 (16²), 289 (17²), 324 (18²), 361 (19²), and 400 (20²). Therefore, the perfect squares in that range are 121, 144, 169, 196, 225, 256, 289, 324, 361, and 400.
They are all perfect squares
The perfect squares between 100 and 600 are 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, and 576. These correspond to the squares of integers from 11 to 24.
If the value applied in the radical is not a perfect square, it is irrational. 25; 400; and 625 are perfect squares and are rational when applied in a radical.
496
20 squares x 20 squares = 400 squares
Ralph likes numbers that are perfect squares but not numbers that are one less than a perfect square. Perfect squares are numbers that can be expressed as the product of an integer multiplied by itself (e.g. 25 = 5 x 5, 400 = 20 x 20, 144 = 12 x 12). Numbers that are one less than a perfect square do not fit this pattern (e.g. 24, 300, 145). Ralph's preference seems to be for the neat, symmetric nature of perfect squares.
361 squares evenly to 19.
John likes numbers that are perfect squares. 400 is 20 squared, 100 is 10 squared, and 3600 is 60 squared. He doesn't like 300, 99, or 3700 because they are not perfect squares. So, there you have it, John has a thing for those perfect squares!