---

Problem 466

---

Distinct terms in a multiplication table

Let P(m,n) be the number of distinct terms in an m×n multiplication table.

For example, a 3×4 multiplication table looks like this:

×	1	2	3	4
1	1	2	3	4
2	2	4	6	8
3	3	6	9	12

There are 8 distinct terms {1,2,3,4,6,8,9,12}, therefore P(3,4) = 8.

You are given that:
P(64,64) = 1263,
P(12,345) = 1998, and
P(32,10¹⁵) = 13826382602124302.

Find P(64,10¹⁶).

---

乘法表中的不同项

记P(m,n)是m×n乘法表中不同的项的个数。

下图是一个3×4的乘法表：

×	1	2	3	4
1	1	2	3	4
2	2	4	6	8
3	3	6	9	12

在这个乘法表中有8个不同的项{1,2,3,4,6,8,9,12}，因此P(3,4) = 8。

已知：
P(64,64) = 1263，
P(12,345) = 1998，以及
P(32,10¹⁵) = 13826382602124302。

求P(64,10¹⁶)。

---