Problem
Multiplication cryptarithm
A two-digit number AB times 6 gives BBB, all three digits the same. Equal letters are equal digits and different letters differ. BBB must be three digits, so B runs 1 to 9. Find A and B.
Reasoning · Grade 3-1 Multiplication
Your answer
Watch out Four BBB values divide by 6, but the quotient must be two digits and end in B — only one survives.
How to solve
Strategy Guess and Check — BBB is one of only nine repdigits (111, 222, ... , 999), and each equals 111 x B. Dividing each by 6 (or checking which are multiples of 6) leaves a tiny set to test, so guessing and checking the few candidate B values pins down the answer.
1STEP 1
Write BBB as 111 x B
All three digits alike makes BBB 111 × B.
BBB = 111 × B
Place value turns the repeated-digit number into one neat product, making the structure obvious.
3.NBT.A.2Make A Systematic ListA three-digit number with all three digits the same is exactly 111 multiplied by that digit.
Why?
The digit is counted a hundred times in the hundreds place, ten times in the tens and once in the ones, and those counts add to a hundred and eleven.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Why?
A hundred of it, ten of it and one of it can be gathered into a single multiplication, because a shared factor may be pulled out of a sum.
🧱Distributive propertya groups of (b and c) is a groups of b plus a groups of c — you can split the group.
2STEP 2
The product must be a multiple of 6
Only 222, 444, 666, 888 divide by 6.
444 = 6 × 74
Checking divisibility by 6 instantly throws out most repdigits, leaving only a couple to test.
3.OA.B.5Guess And Check3STEP 3
Match AB to the repdigit
Only 444 = 74 × 6 has a two-digit quotient ending in B.
74 × 6 = 444, A = 7, B = 4
Only one candidate makes every letter match the same digit consistently, so the puzzle has a unique answer.
3.OA.A.4Guess And CheckAnswer
A = 7, B = 4 (74 × 6 = 444)
74 x 6 = 444, a three-digit number with all equal digits, and its repeated digit 4 matches the ones digit of 74, exactly as the BBB pattern requires. A = 7 and B = 4 are different digits, satisfying the cryptarithm rule.
Takeaway
A number like 444 is just 111 four times, so Grade 3 place-value sense cracks the whole puzzle!
- Write BBB as 111 x B
- The product must be a multiple of 6
- Match AB to the repdigit