Why use hex?

In both cases you cite, the bit pattern of the number is important, not the actual number.

For example,In the first case, j is going to be 1, then 2, 4, 8, 16, 32, 64 and finally 128 as the loop progresses.

In binary, that is,

0000:0001, 0000:0010, 0000:0100, 0000:1000, 0001:0000, 0010:0000, 0100:0000 and 1000:0000.

There's no option for binary constants in C or C++, but it's a bit clearer in Hex: 0x01, 0x02, 0x04, 0x08, 0x10, 0x20, 0x40, and 0x80.

In the second example,the goal was to remove the lower two bytes of the value. So given a value of 1,234,567,890 we want to end up with 1,234,567,168.
In hex, it's clearer: start with 0x4996:02d2, end with 0x4996:0000.

its a bit mask. Hex values make it easy to see the underlying binary representation. n & 0xffff0000 returns the top 16 bits of n. 0xffff0000 means "16 1s and 16 0s in binary"

0x80 means "1000000", so you start with "00000001" and continue shifting that bit over to the left "0000010", "0000100", etc until "1000000"

There's a direct mapping between hex (or octal for that matter) digits and the underlying bit patterns, which is not the case with decimal. A decimal '9' represents something different with respect to bit patterns depending on what column it is in and what numbers surround it - it doesn't have a direct relationship to a bit pattern. In hex, a '9' always means '1001', no matter which column. 9 = '1001', 95 = '*1001*0101' and so forth.

As a vestige of my 8-bit days I find hex a convenient shorthand for anything binary. Bit twiddling is a dying skill. Once (about 10 years ago) I saw a third year networking paper at university where only 10% (5 out of 50 or so) of the people in the class could calculate a bit-mask.