Problem
Find a number from layered conditions
Look for a seven-digit number. Its millions digit is 5 more than its ones digit. The digit 0 appears exactly three times. Find the largest such number.
Reasoning · Grade 4-1 Large Numbers
Your answer
Watch out Adding more zeros is tempting, but there must be exactly three — a fourth breaks the condition.
How to solve
Strategy Eliminate Possibilities — There are millions of seven-digit numbers, so the conditions have to do the searching rather than a list. I apply the condition that pins down the most first — the one linking the millions digit to the ones digit — because the millions place decides size more than any other. Once the two end digits are fixed, only five places remain, and the three zeros can be placed by simple reasoning about where a 0 costs the least.
1STEP 1
Name the seven places
Left places weigh most, so start at the millions digit.
Knowing which place carries the most weight tells me which digit to make as large as possible first.
4.NBT.A.2Analyze The Units2STEP 2
Push the millions digit as high as the conditions allow
With 9 in the millions place, the ones digit is 4.
9 - 4 = 5
Because the leading place outweighs all the others put together, fixing it at its maximum settles the biggest part of the answer before anything else is decided.
4.NBT.A.2Eliminate PossibilitiesFixing the millions digit as high as the conditions allow settles the biggest part of the answer before anything else is decided.
Why?
A digit in the millions place counts for a million of itself, while all the places below it together can never reach one million.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Why?
The number is what all its place amounts come to together, so raising the heaviest place lifts it further than any change below could.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
3STEP 3
Work out where the three zeros must go
Since 9 and 4 are non-zero, the zeros fall in the five middle places.
Fixing the two end digits first turned a seven-place puzzle into a five-place one, small enough to reason through directly.
4.NBT.A.2Eliminate Possibilities4STEP 4
Place the two non-zero digits as far left as possible
Fill the two free places on the left with 9 and 9.
9 990 004
Comparing two numbers means comparing them place by place from the left, so putting the big digits leftmost and the forced zeros rightmost is exactly what makes the number largest.
4.NBT.A.2Make A Systematic List5STEP 5
Confirm no larger number exists
So the largest number is 9990004.
Checking that every way of making the number bigger breaks one of the conditions turns a good candidate into a proven answer.
4.NBT.A.2Eliminate PossibilitiesAnswer
9990004
9 − 4 = 5
The answer has exactly seven digits; its millions digit 9 is 5 more than its ones digit 4; and its digits 9, 9, 9, 0, 0, 0, 4 contain exactly three zeros — all three conditions hold. It also sits just under ten million, which is where the largest seven-digit numbers live, so the size is sensible.
Takeaway
To build the biggest number, spend your best digits on the leftmost places first — that is the same place-value comparison you already use to tell which number is bigger.
- Name the seven places
- Push the millions digit as high as the conditions allow
- Work out where the three zeros must go
- Place the two non-zero digits as far left as possible
- Confirm no larger number exists