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#### Number of digits in products

Product of integers with N and M digits
N = 2, M = 3: 10 * 100   =   1000,   D = 4 = [N + M - 1]
N = 2, M = 3: 99 * 100   =   9900,   D = 4 = [N + M - 1]
N = 2, M = 3: 10 * 999   =   9990,   D = 4 = [N + M - 1]
N = 2, M = 3: 99 * 999   = 98901,   D = 5 = [N + M]

N = 2, M = 4: 10 * 1000 =   10000, D = 5 = [N + M - 1]
N = 2, M = 4: 99 * 1000 =   99000, D = 5 = [N + M - 1]
N = 2, M = 4: 10 * 9999 =   99990, D = 5 = [N + M - 1]
N = 2, M = 4: 99 * 9999 = 989901, D = 6 = [N + M]
Thus: the answer is N+M-1 and N+M.

#### Last digit of power of 3, 4, 7 ...

Note:
3^4 = 81
3^5 = 243
3^6 = 729
3^7 = 2187
3^8 = 6561
Thus:
3^n = 1 if remainder(n/4) = 0
= 3 if remainder(n/4) = 1
= 9 if remainder(n/4) = 2
= 7 if remainder(n/4) = 3
Similarly, last digit of 4^n is 6 is n is an even number and 4 if n is an odd number.

Similarly, last digit of 7^n

Thus:
7^n = 1 if remainder(n/4) = 0
= 7 if remainder(n/4) = 1
= 9 if remainder(n/4) = 2
= 3 if remainder(n/4) = 3