Depth · one concept from first principles · step 2 of 4
Product and sum signs fix how many are negative
Three integers with strictly increasing absolute values add to -7 and multiply to 12. Find them.
Number systemOperations
Your answer
How to solve
Strategy Identify Subproblems — Separate the signs from the sizes. The signs of the product and the sum decide how many of the numbers are negative; the absolute values must be three different whole numbers multiplying to 12, of which there are only two choices. Then the sum picks the one that works.
1STEP 1
Count the negatives
A positive product has 0 or 2 negative factors. With none, the sum would be positive, not -7, so exactly two are negative.
abc = 12 ＞ 0, a + b + c = -7 ＜ 0
Each negative factor flips the sign of the product.
7.NS.A.2Identify SubproblemsExactly two of , , are negative and one is positive.
Why?
Each negative factor flips the sign of the product and two flips land back on positive, so a positive product like needs an even number of negatives: or .
🧱Signs of products and quotientsMultiplying by a negative number flips a number to the other side of zero, so two flips land back on the positive side: for two nonzero numbers, a product or quotient with like signs is positive and one with unlike signs is negative (zero itself has no sign).
Why?
With no negatives the sum of three positive integers would be positive, not , so that option is ruled out and two negatives is the one left.
🧱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 the possible sizes
|a|, |b|, |c| are three different positive integers in increasing order with product 12.
|a| × |b| × |c| = 12 = 1 × 2 × 6 = 1 × 3 × 4
Only {1, 2, 6} and {1, 3, 4} work; 1 × 1 × 12 and 2 × 2 × 3 repeat a value.
4.OA.B.4Make A Systematic List3STEP 3
Write the sum in terms of the positive one
Let p be the size of the positive number and S the total of all three sizes. The other two are negative, so the sum is p - (S - p).
p - (S - p) = 2p - S = -7
Subtracting the two negative sizes from the positive one gives the sum.
7.NS.A.1Convert To Algebra4STEP 4
Test both size sets
For {1, 2, 6}, S = 9 and p = 1. For {1, 3, 4}, S = 8 would need p = 1/2, which is not an integer.
S = 9: 2p - 9 = -7, p = 1; S = 8: 2p - 8 = -7, p = 1/2
So the positive number has size 1 and the negatives have sizes 2 and 6: a = 1, b = -2, c = -6.
7.NS.A.1Make A Systematic ListAnswer
a=1, b=-2, c=-6
|1| ＜ |-2| ＜ |-6|, 1 + (-2) + (-6) = -7, and 1 × (-2) × (-6) = 12. All three conditions hold.
Takeaway
Settle the signs first, then the sizes: two short lists beat one long search.
- Count the negatives
- List the possible sizes
- Write the sum in terms of the positive one
- Test both size sets
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