Are all integers with exponent over 52 are even in 64 bit floating point

Question

Am I correct to conclude that:

• all integers with exponent over 52 in the 64 bit floating point will be even? For example, if exponent is 53, and the mantissa is 0000000000000000000000000000000000000000000000000001, then the number is 100000000000000000000000000000000000000000000000000010 - ends in 10. If the mantissa is 54 the number ends with 100.

• the more we increase the exponent, the more integer numbers can not be represented - for the exponent of 54 - it's impossible to represent one number - 9007199254740993, with exponent of 55 - we can't represent 3 numbers (because we have two bits left out of mantissa which can have combinations of 01,10,11).