Problem 23 · 2015 Math Kangaroo
Medium
Arithmetic & Operations
groupingarithmetic-series
The following table is the multiplication table of the numbers 1 to 10. What is the sum of all 100 products in the complete table?
| × | 1 | 2 | 3 | … | 10 |
|---|---|---|---|---|---|
| 1 | 1 | 2 | 3 | … | 10 |
| 2 | 2 | 4 | 6 | … | 20 |
| 3 | 3 | 6 | 9 | … | 30 |
| … | … | … | … | … | … |
| 10 | 10 | 20 | 30 | … | 100 |
Show answer
Answer: D — 3025
Show hints
Hint 1 of 2
Group the 100 products by which row they are in.
Still stuck? Show hint 2 →
Hint 2 of 2
Each row is one number times (1+2+…+10).
Show solution
Approach: factor the table sum as a product of two row sums
- Row k of the table sums to k×(1+2+…+10) = k×55.
- Adding all rows gives (1+2+…+10)×55 = 55×55 = 55².
- So the total of all 100 products is 3025 (D).
Mark:
· log in to save