What is the maximum length of the multiplication of two integers?

Question

Given two non-negative integers m1 and n1, m1 has m digits and n1 has n digits. p1 is the product of m1 and n1 (p1 = m1 x n1), p1 has p digits.

What is the maximum value of p?

Answer

• 0 ≤ m1 ≤ 10^m - 1
• 0 ≤ n1 ≤ 10ⁿ - 1
• 10^k - 1 is an integer has k digits of 9

10^k - 1 = 9.....9
           |_____|
        k digits of 9
10^k - 1 = 9.....9
           |_____|
        k digits of 9

p1 = m1 x n1 ≤ (10^m -1) x (10ⁿ -1)
   = 10^m+n - 10^m - 10^m + 1
   = 10^m+n - 1 - (10^m - 1) - (10^m - 1)
   ≤ 10^m+n - 1

So, p1 can have maximum m + n digits.