Competition · AMC preparation · step 4 of 4
AMC 8 · 2001 · #4
Grade 4 number-theoryarithmeticPick an answer.
AMC 8 2001 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #5 (Look for a Pattern) gives the place-value rule: in a multi-digit number, the leftmost place matters most, so to keep the number small we should put the smallest available digit on the left first, then the next smallest, and so on. Tool #6 (Guess and Check) handles the twist — the even-number rule forces the units digit to be 2 or 4, so the greedy left-to-right plan can run into a conflict when it reaches the last two places. When that happens, we swap the units digit and try again.
Make the front digits small
Greedy left-to-right: place the smallest available digit first, which gives 12349.
Place value says the ten-thousands slot is worth the most, so the smallest digit goes there; then the thousands slot, and so on.
4.NBT.A.2Look For A PatternCheck the even rule
12349 ends in an odd digit, so it fails — an even number needs its units digit to be 2 or 4.
A number is even exactly when its units digit is even, and the only even digits available are 2 and 4.
2.OA.C.3Guess And CheckFix the units digit
Put the larger even digit 4 in the units place and fill the rest smallest-first, giving 12394.
Sending 4 (rather than 2) to the units place is better because 2 is smaller and helps more in the higher places.
To make the number as small as possible, the larger even digit 4 should sit in the units place and the smaller even digit 2 should move to a higher place.
▸ Why?
The number must be even, so its units place is forced to hold an even digit, and 2 or 4 are the only even digits available.
▸ Why?
A whole number is a count of tens plus its units digit, and every ten is itself even, so the tens part always splits into two equal groups on its own — which leaves the units digit as the only thing that can decide whether the whole number splits evenly.
▸ Why?
One of the two even digits is stuck in the low units place, so giving up the larger one (4) there leaves the smaller one (2) free to take a higher place, where a small digit shrinks the number the most.
▸ Why?
Each place to the left is worth ten of the place on its right, so a digit in a higher place counts for far more; putting the smaller digit where the weight is larger lowers the total more than putting it where the weight is small.
Read off the tens digit
The tens digit of 12394 is the second digit from the right: 9.
The tens place is the second digit from the right, which is 9.
4.NBT.A.2Look For A PatternTo shrink a number, push the smallest digits to the leftmost places — but if a rule like "must be even" claims a spot, give that spot the larger acceptable digit so the smaller ones can do more work on the left.
- Make the front digits small
- Check the even rule
- Fix the units digit
- Read off the tens digit
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