AMC 10 · 2015 · #6
Grade 5 rate-ratioPick an answer.
Nobody wants to compute 169 products. The question only asks whether each entry is odd, and whether a product is odd depends on nothing but the parity of the two labels. So the plan is: first pin down exactly how many cells the body has, then prove the parity rule for products in both directions, then count the odd labels and multiply. Because the whole result rests on two claims that are easy to state and easy to get wrong — that the body has 13 rows and not 12, and that a product is odd exactly when both factors are odd — each gets its own proof rather than being assumed. A complement count and a size bound on the answer choices then check the result along two routes that do not reuse the same arithmetic.
Count the cells in the body
The body holds 169 cells.
A list that starts at 0 and ends at 12 holds 13 numbers, so the body is one row and one column bigger than the name "12 times table" makes it sound.
5.NBT.B.5Draw A DiagramProve when a product is odd
A product is odd only if both labels are.
A single factor of 2 anywhere in a product survives to the end, and if no factor carries one, nothing can create it.
A single factor of two anywhere in a product survives to the end, and if no factor carries one, nothing can create it.
▸ Why?
Every number has exactly one prime recipe, so the twos in a product come only from its factors.
▸ Why?
A product is odd exactly when every factor is odd, since one even factor is enough to make it even.
List the odd labels
There are six odd labels.
In a run of consecutive whole numbers starting and ending on an even number, the evens outnumber the odds by exactly one.
2.OA.C.3Make A Systematic ListCount the odd cells
That gives 36 odd cells.
Odd rows crossed with odd columns is itself a multiplication table, so the odd cells are a 6 by 6 block hiding inside the 13 by 13 one.
3.OA.A.1Eliminate PossibilitiesCheck by counting the evens
Counting the evens confirms it.
Counting what you do not want and subtracting is a genuinely different computation, so agreement between the two is real evidence and not just the same slip repeated.
4.NBT.B.4Change Focus Count The ComplementRound the fraction without dividing
Bracketing rounds it to 0.21.
To round a fraction you never need its digits, only which side of the two nearest halfway marks it falls on.
5.NBT.A.4Solve An Easier Related ProblemConfirm against the choices
It is safely below a quarter, choice (E).
Adding the even label 0 stretches the denominator without adding a single odd entry, so it can only drag the share down.
4.NF.A.2Eliminate PossibilitiesA product is odd only when every factor is odd, so counting odd entries in a times table is just counting odd row labels times odd column labels — and the row and column for 0 still count as part of the table.
- Count the cells in the body
- Prove when a product is odd
- List the odd labels
- Count the odd cells
- Check by counting the evens
- Round the fraction without dividing
- Confirm against the choices