Reasoning · Grade 2-2 The Multiplication Table

Problem

Units-digit pattern of each times table

For each times table 1 to 9, count how many different ones digits its multiples 1× to 9× show. Count the kinds of digit, not the number of multiples. The 2 table is the given example in the '5 digits' box. Sort all nine tables into the 2, 5 and 9 boxes.
Your answer
How to solve
Strategy Make a Systematic List — List the ones digits of each times table's multiples and count the different ones. A pattern quickly emerges: the answer depends on whether the table number is 5 (ends in 5 or 0), even (ends in an even digit), or odd-and-not-5, so we can group all nine tables by that.
1STEP 1

List the ones digits of a few tables

The 5 table cycles 5 and 0; the 2 table cycles 0, 2, 4, 6, 8.

5 table→{0,5}; 2 table→{0,2,4,6,8}; 3 table→{1,…,9}
2STEP 2

See the pattern behind the counts

An even step gives 5, a step of 5 gives 2, any other odd step gives 9.

by 5→ 2; even step→ 5; odd ≠ 5→ 9
3STEP 3

Sort all nine tables

So 2 is the 5 table, 5 is 2·4·6·8, and 9 is 1·3·7·9.

2→{5}; 5→{2,4,6,8}; 9→{1,3,7,9}
Answer
2 digits = the 5 table / 5 digits = 2, 4, 6, 8 / 9 digits = 1, 3, 7, 9
All nine tables are sorted exactly once: 1 table in the '2 digits' box, 4 tables in the '5 digits' box, and 4 tables in the '9 digits' box, totaling 1 + 4 + 4 = 9. The given example (2 table = 5 digits) lands in the '5 digits' box as expected, confirming the rule.
Takeaway

This only needs Grade 3 skip-counting: the last digits repeat in a cycle, and how long that cycle is tells you 2, 5, or 9 different digits!

  • List the ones digits of a few tables
  • See the pattern behind the counts
  • Sort all nine tables