Problem
Collapse a repeated number or expression
Work out 2020 × 20212021 − 2021 × 20202020. Each eight-digit number is a four-digit block written twice. Multiplying by hand is slow and error-prone. Find the value of the expression.
Reasoning · Grade 6-2 Mixed Operations with Fractions
Your answer
Watch out The two products look different, but they are the same product reordered.
How to solve
Strategy Solve an Easier Related Problem — Eight-digit numbers are unreadable, so I start with the smallest number that has the same shape: a two-digit block written twice, like 2424. That tiny case shows exactly where the repeated-block factor comes from and why it is a string of 1s and 0s, and the reason transfers to a four-digit block without any new idea. Then I rewrite both eight-digit numbers as products, and the pattern in the rewritten expression — the same three factors on both sides of the minus sign — is what finishes the problem. Nothing bigger than 2021 x 2 ever gets multiplied.
1STEP 1
Try the same shape with a two-digit block: what is 2424 really?
Notice first that 2424 is 24 × 101.
2424 = 24 × 100 + 24 × 1 = 24 × (100+1) = 24 × 101
Writing a block twice is the same as making one copy and sliding it left past the other, and sliding a number left in our place-value system multiplies it by a power of 10. That is why a repeat factor always looks like 1 followed by zeros followed by 1.
5.NBT.A.1Solve An Easier Related ProblemA number like 2424 is really the block 24 multiplied by 101, because the block sits in two places at once.
Why?
Sliding a block two places to the left multiplies it by a hundred, so the two copies are worth a hundred of it and one of it.
🧱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 plus one of it can be gathered into a single multiplication, because a shared factor lifts 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
Slide the block four places instead of two
Slid four places, each is the block × 10001.
20212021 = 2021 × 10000 + 2021 = 2021 × 10001, 20202020 = 2020 × 10001
The pattern is one rule with a dial on it: a 2-digit block repeated gives x 101, a 3-digit block gives x 1001, a 4-digit block gives x 10001. The number of zeros in the middle is just how far the copy had to slide, so you can write it down by counting digits.
5.NBT.A.1Look For A Pattern3STEP 3
Rewrite the whole expression using the two factorisations
Rewrite the whole expression that way.
2020 × 20212021 - 2021 × 20202020 = 2020 × (2021 × 10001) - 2021 × (2020 × 10001)
Swapping a number for an equal product is always allowed, and it is worth doing here because a product shows its ingredients while an eight-digit numeral hides them.
6.EE.A.3Organize Information In More Ways4STEP 4
See that the two terms are the same product written in a different order
The two terms become the same product.
2020 × 2021 × 10001 - 2021 × 2020 × 10001 = 0
You never learn what the common product is, and you do not need to: once both sides of the minus sign are the same collection of factors, the answer is 0 no matter how big that collection multiplies out to.
6.EE.A.3Look For A Pattern5STEP 5
Confirm with a small copy of the same puzzle
So the value is 0.
24 × 2525 - 25 × 2424 = 60600 - 60600 = 0
The small copy can be finished by ordinary multiplication in under a minute, so it is a real test of the shortcut rather than a restatement of it.
5.NBT.B.5Solve An Easier Related ProblemAnswer
0
20212021 = 2021 × 10001
The two products are enormous — each is a bit over 4 x 10¹0 — so an answer of 0 is only believable if they are exactly equal, and they are: both equal 2020 x 2021 x 10001, the same three factors in a different order. A quick size check also agrees that they must be close: 2020 x 20212021 and 2021 x 20202020 both have one factor about 2020 and one about 20.2 million, and the tiny advantages cancel (the first term has the bigger eight-digit number, the second has the bigger multiplier). Finally, the answer should be a whole number with no units attached, since the problem is pure arithmetic on whole numbers, and 0 is.
Takeaway
A number that repeats a block is that block times 10001 (or 101, or 1001) — pull it out and the two giant products turn out to be twins.
- Try the same shape with a two-digit block: what is 2424 really?
- Slide the block four places instead of two
- Rewrite the whole expression using the two factorisations
- See that the two terms are the same product written in a different order
- Confirm with a small copy of the same puzzle