Problem
Find a Number Meeting Digit Conditions
A two-digit number has its digits swapped. The swap makes it larger, so the original's ones digit beats its tens digit. The two numbers multiply to 2944. Find the original number.
Reasoning · Grade 3-2 Cryptarithms and Hidden-Digit Puzzles
Your answer
Watch out Finding 46 and 64 feels done, but the swap made it larger, so the original is the smaller one.
How to solve
Strategy Guess and Check — Estimate roughly where the two numbers are (both near the number that times itself makes about 2944), which makes the guesses small in count, then guess two-digit numbers and their digit-swaps and check the product until it hits 2944.
1STEP 1
Estimate the size of the two numbers
A swapped pair sits close together, so both are near 50.
50 × 50 = 2500, 60 × 60 = 3600, 2500 ＜ 2944 ＜ 3600
Squaring nearby round numbers brackets the answer, so I only need to try a few two-digit numbers, not all of them.
3.NBT.A.1Solve An Easier Related ProblemSquaring a few round numbers brackets the answer, so only a handful of two-digit numbers have to be tried.
Why?
A bigger number makes a bigger product, so once a round number overshoots, everything above it overshoots too.
🧱Transitivity of orderIf the first is smaller than the second and the second smaller than the third, the first is smaller than the third, so pairwise comparisons chain into one full order.
Why?
Everything outside the bracket is ruled out in one stroke, so the search starts from a short list instead of every two-digit number.
🧱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
List digit-swap pairs near that size where the swap is larger
2944 ends in 4, and only 4 and 6 multiply to an ending 4.
4 × 6 = 24 (ends in 4)
Checking the ones digit of the product is a fast filter that singles out the pair 46 and 64.
3.OA.C.7Make A Systematic List3STEP 3
Check 46 × 64
Checking, 46 × 64 = 2944 exactly.
46 × 64 = 2760 + 184 = 2944
Breaking the multiplication into 46×60 and 46×4 keeps it to easy parts that add to the target.
4.NBT.B.5Guess And Check4STEP 4
Confirm it is the original number
The swap made it larger, so the original is 46.
46 ＜ 64, original = 46
The 'swap is larger' clue tells you which of the two numbers was the starting one.
4.NBT.A.2Guess And CheckAnswer
46
46 × 64 = 2944
Check all clues: 46 is two digits; swapping gives 64, which is larger than 46 (yes); and 46 × 64 = 2944 (yes). Everything matches, so 46 is correct.
Takeaway
A good estimate plus the ones-digit clue narrows two-digit guessing down to just one pair — the original number is 46!
- Estimate the size of the two numbers
- List digit-swap pairs near that size where the swap is larger
- Check 46 × 64
- Confirm it is the original number