Problem
Complete the multiplication equation
A two-digit times a one-digit must land on 7□6. The product lies from 706 to 796 and ends in 6. The two-digit factor is 10 to 99, the one-digit 1 to 9. Complete three different equations.
Reasoning · Grade 3-1 Multiplication
Your answer
Watch out Any one-digit factor seems worth trying, but with 99 as the cap only 8 or 9 can pass 700.
How to solve
Strategy Make a Systematic List — The product must lie in the small range 706-796 and end in 6, so the candidate two-digit x one-digit pairs form a tiny finite set. Listing the multiples of each one-digit number that land in 706-796 and keeping those ending in 6 finds every valid equation.
1STEP 1
Bound the product
7□6 puts the product between 706 and 796, ending in 6.
706 ≤ product ≤ 796
Knowing the hundreds and ones digits pins the product into a range of only about 90 numbers - small enough to search.
3.OA.A.4Make A Systematic ListKnowing the hundreds digit and the ones digit traps the product inside a range of about ninety numbers.
Why?
The hundreds digit fixes which hundred the product lies in, and the ones digit fixes where inside that hundred it can land.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Why?
Every product outside that range is thrown out at once, so only the handful left has to be tested by hand.
🧱Rule out the rest and the last one is forcedIf every possibility but one has been ruled out, the survivor must be the answer — no further checking is needed.
2STEP 2
Search products ending in 6
Only a one-digit factor of 8 or 9 reaches that range, giving 84 × 9, 92 × 8, 97 × 8.
84 × 9 = 756, 92 × 8 = 736, 97 × 8 = 776
Multiplying within 100 is fluent grade-3 work, so checking a handful of candidates is quick and certain.
3.OA.C.7Guess And Check3STEP 3
Read off the tens digit
Reading the tens digit off each gives 5, 3, 7.
7 5 6, 7 3 6, 7 7 6
Reading the digits of the product is just place-value identification.
3.NBT.A.2Make A Systematic ListAnswer
84 × 9 = 756, 92 × 8 = 736, 97 × 8 = 776
Each product is between 706 and 796, starts with 7, and ends with 6, matching 7□6. A systematic sweep of multipliers 1-9 turns up no other two-digit factor whose product fits, so exactly three completions exist.
Takeaway
Knowing just the first and last digit of the answer lets you hunt down every equation with Grade 3 times tables!
- Bound the product
- Search products ending in 6
- Read off the tens digit