Alright, settle in. Let’s talk about the number 1260. Ever look at a number like that and just… wonder? It’s not tiny, not huge. Feels substantial, doesn’t it? And then that classic brain-tickler pops up, the kind you might’ve mumbled to yourself during a long commute or maybe saw scrawled on a whiteboard somewhere: 几乘几等于1260？ Simple question on the surface, right? But peeling back the layers? Ah, that’s where the fun is. It’s not just one answer, you see. This number, 1260，它肚子里藏着不少秘密，等着咱们去敲门、去发现。

Think of it like this: 1260 is a city. And the question “几乘几等于1260” is asking for all the different pairs of roads, say, Road A and Road B, that meet up exactly at the city center, summing up, or rather, multiplying up, to this precise location. Every single pair is a valid route. How do we find all these routes?

Forget just guessing. That’s like wandering around hoping to stumble upon every single street corner. INEFFICIENT. The real key, the map to this city of 1260, is something they drummed into our heads in school, maybe a bit dryly back then, but it’s pure magic now: 质因数分解。 Breaking a number down into its absolute, indivisible building blocks. Like finding the atoms that make up the molecule of 1260.

Let’s do it together, shall we? 1260. Well, it ends in zero, so it’s definitely friends with 10. And 10 is 2 times 5. So, 1260 = 126 * 10 = 126 * 2 * 5. Okay, 126. Ends in 6, so it likes 2. 126 divided by 2? That’s 63. Aha, 63! That number smells of threes and sevens. 63 = 9 * 7. And 9? That’s 3 * 3.

Putting it all back together: 1260 = 126 * 2 * 5 = (2 * 63) * 2 * 5 = (2 * 9 * 7) * 2 * 5 = (2 * (3 * 3) * 7) * 2 * 5. Rearranging the blocks, gathering identical ones: 1260 = 2 * 2 * 3 * 3 * 5 * 7. Or, more neatly, using exponents: 1260 = 2² × 3² × 5¹ × 7¹。

See? These are the atoms: two ‘2’s, two ‘3’s, one ‘5’, one ‘7’. Any number that divides 1260, any factor, is just a combination of these atoms. You can pick some 2s (zero, one, or two of them), some 3s (zero, one, or two), some 5s (zero or one), and some 7s (zero or one).

Now, the question isn’t just about factors, it’s about pairs that multiply to 1260. If one number in the pair is ‘a’, the other must be ‘1260 divided by a’. So, finding all the pairs is exactly the same as finding all the factors and then just matching them up. For every factor ‘a’ you find, its partner ‘b’ is automatically determined.

Let’s list out the factors systematically, building them from our prime atoms. It’s like setting up all the possible routes leaving the city center.
Start simple:
* Take none of the atoms (well, technically take 2⁰, 3⁰, 5⁰, 7⁰, which is 1). Factor is 1. Partner? 1260 / 1 = 1260. Pair: (1, 1260). Obvious, but counts!
* Take just one type of atom:
* One 2: Factor 2. Partner 1260/2 = 630. Pair: (2, 630).
* One 3: Factor 3. Partner 1260/3 = 420. Pair: (3, 420).
* One 5: Factor 5. Partner 1260/5 = 252. Pair: (5, 252).
* One 7: Factor 7. Partner 1260/7 = 180. Pair: (7, 180).
* Take two of the same atom, or combinations of two different types:
* Two 2s: 2² = 4. Partner 1260/4 = 315. Pair: (4, 315).
* Two 3s: 3² = 9. Partner 1260/9 = 140. Pair: (9, 140).
* 2 * 3 = 6. Partner 1260/6 = 210. Pair: (6, 210).
* 2 * 5 = 10. Partner 1260/10 = 126. Pair: (10, 126).
* 2 * 7 = 14. Partner 1260/14 = 90. Pair: (14, 90).
* 3 * 5 = 15. Partner 1260/15 = 84. Pair: (15, 84).
* 3 * 7 = 21. Partner 1260/21 = 60. Pair: (21, 60).
* 5 * 7 = 35. Partner 1260/35 = 36. Pair: (35, 36).

• Combinations of three atoms (or more):

  • 2² * 3 = 4 * 3 = 12. Partner 1260/12 = 105. Pair: (12, 105).
  • 2² * 5 = 4 * 5 = 20. Partner 1260/20 = 63. Pair: (20, 63).
  • 2² * 7 = 4 * 7 = 28. Partner 1260/28 = 45. Pair: (28, 45).
  • 2 * 3² = 2 * 9 = 18. Partner 1260/18 = 70. Pair: (18, 70).
  • 2 * 3 * 5 = 30. Partner 1260/30 = 42. Pair: (30, 42).
  • 2 * 3 * 7 = 42. (Hey, we found its partner already! This just confirms our list is working).
  • 2 * 5 * 7 = 70. (Again, partner found).
  • 3² * 5 = 9 * 5 = 45. (Partner found).
  • 3² * 7 = 9 * 7 = 63. (Partner found).
  • 3 * 5 * 7 = 105. (Partner found).

• Combinations of four atoms:

  • 2² * 3² = 4 * 9 = 36. (Partner found).
  • 2² * 3 * 5 = 60. (Partner found).
  • 2² * 3 * 7 = 84. (Partner found).
  • 2 * 3² * 5 = 90. (Partner found).
  • 2 * 3² * 7 = 126. (Partner found).
  • 2 * 3 * 5 * 7 = 210. (Partner found).
  • 3² * 5 * 7 = 315. (Partner found).

• Combinations of five atoms:

  • 2² * 3² * 5 = 180. (Partner found).
  • 2² * 3² * 7 = 252. (Partner found).
  • 2² * 3 * 5 * 7 = 420. (Partner found).
  • 2 * 3² * 5 * 7 = 630. (Partner found).

• All six atoms: 2² * 3² * 5 * 7 = 1260. Partner 1260/1260 = 1. (Partner found right at the start!).

Phew! Listing them out like that, systematically building from the prime factors, gives you every single factor. And for every factor, there’s a partner waiting. The total number of factors, by the way, is calculated from the exponents of the prime factors: (2+1) * (2+1) * (1+1) * (1+1) = 3 * 3 * 2 * 2 = 36 factors. Since 1260 isn’t a perfect square (because the exponents in its prime factorization are not all even), none of the factors is multiplied by itself to get 1260. So, these 36 factors pair up neatly into 36 / 2 = 18 unique pairs.

So, what are these pairs? The full list of answers to 几乘几等于1260 is:
1 × 1260
2 × 630
3 × 420
4 × 315
5 × 252
6 × 210
7 × 180
9 × 140
10 × 126
12 × 105
14 × 90
15 × 84
18 × 70
20 × 63
21 × 60
28 × 45
30 × 42
35 × 36

Eighteen different ways to multiply two whole numbers and land exactly on 1260. Isn’t that something? It shows how one number can be built from different sets of foundations. It’s not just one size fits all. A small number and a large one, two medium-sized ones, pairs where one is even and one is odd (except when the factor includes a 2!), pairs where both are even… a whole spectrum of possibilities.

For me, tackling a question like 几乘几等于1260 is a little reminder of how interconnected numbers are. They aren’t just abstract symbols. They have relationships, structures, even family trees (those prime factors!). And figuring out these relationships, like finding all the pairs that make 1260, feels like solving a small, satisfying puzzle the universe left lying around. It’s more than just arithmetic; it’s about understanding the architecture of numbers. Each pair, be it the obvious (1, 1260) or the less intuitive (35, 36), tells a little story about 1260, a different path to the same destination. And knowing how to find them all, using 质因数分解 as our map, that feels like real power. Not the “rule the world” kind, but the quiet, “I understand something fundamental” kind. Next time you see 1260, maybe you won’t just see a number, but a number with 18 different identities hiding within its multiplication pairs. Pretty neat, huh?