Here is how to get 15.

The 1 table is transparent.

If you know addition, the 2 table is just adding a number to itself.

For the 5 table, think of dividing by 2 and then multiplying by 10. 5×7 = 7/2×10 = 3.5×10 = 35.

There are two tricks that can be used to multiply by 9. You can multiply by 10 and subtract. 9×7 = 10×7 – 7 = 70 – 7 = 63. Alternatively, use the rule that the first digit is one less than the multiplier and realize the two digits add to 9. So for 9×7, 6 is one less than 7 and 6+3=9.

That leaves 5 numbers – 3, 4, 6, 7 and 8. A 5 by 5 table has 25 entries. 5 of these are along the diagonal – 3×3, 4×4, etc. The remaining ones occur twice – 3×7 and 7×3. It is only necessary to memorize one of these, eliminating (25 – 5)/2 = 10. 25 – 10 =15 entries to memorize. Over the course of a year, that does not seem like all that much.