Modulo operation: Why does -1 = 59 (mod 60), whereas 1 = 1(mod 60)? The latter makes perfect sense to me, but what's the skinny with the former?  I researched but couldn't understand the drift. TIA
Consider the Number line marked with integers. Now, Paste the integer $60$ over $0$ and wrap around the entire number line over the circle formed as above.

Now, we see that when we count on the above number line constructed by the above procedure,  the integer $-1$ coincides with the integer $59$. This is called Modular Arithmetic.

Remarks:
$1$. The above construction can be generalized to any positive integer $n$ by replacing $60$ with $n$.
$2$. In general, while doing Modular arithmetic, we look for a solution inside the set $\{0,1,2,...,n-1\}$. But we can consider any set containing "$n$" consecutive integers.

Note: You can refer to any Elementary Number Theory Book. It will be more helpful.

This video Link might be useful for the above-said construction

I hope this answers your questions.
See here:


  The % (modulo) operator yields the remainder from the division of the first argument by the second. The numeric arguments are first converted to a common type. A zero right argument raises the ZeroDivisionError exception. The arguments may be floating point numbers, e.g., 3.14%0.7 equals 0.34 (since 3.14 equals 4*0.7 + 0.34.) The modulo operator always yields a result with the same sign as its second operand (or zero); the absolute value of the result is strictly smaller than the absolute value of the second operand [2].


So -1 % 60 must return a nonnegative (the same sign as 60) integer that is less than 60 in absolute value, and is congruent to -1 modulo 60. The unique integer satisfying that is 59. If you instead did -1 % -60 you would get -1.