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

Thu, 06 Oct 2022 15:43:15 GMT

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 ≤ 10m - 1

0 ≤ n1 ≤ 10n - 1

10k - 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 ≤ (10m -1) x (10n -1)
= 10m+n - 10m - 10m + 1
= 10m+n - 1 - (10m - 1) - (10m - 1)
≤ 10m+n - 1

So, p1 can have maximum m + n digits.