What is the smallest integer with exactly 100 factors?

Question

Include 1 and integer as two of the factors.

This is not a homework problem. I just feel like giving 10 points to some deserving soul.

Don't worry, Scott, you're getting the points. My question certainly doesn't imply only prime factors, first because I didn't say "prime factors" and secondly because I said the integer is one of the factors. So there! :)

Answer 1

How about 45360 = 2^4 * 3^4 * 5 * 7
# factors = (4+1)(4+1)(1+1)(1+1) = 5*5*2*2 = 100
The number of factors of the number:
x = p(1)^r(1) * p(2)^r(2) * p(3)^r(3) * ... * p(n)^r(n), where this is the prime factorization, is:
(r(1) + 1)(r(2) + 1)(r(3) + 1)...(r(n) + 1)

The question did not ask for just prime factors but all factors.
For instance, if 2 and 3 are factors then so is 6.
45360 has exactly 100 factors.
They are:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 27, 28, 30, 35, 36, 40, 42, 45, 48, 54, 56, 60, 63, 70, 72, 80, 81, 84, 90, 105, 108, 112, 120, 126, 135, 140, 144, 162, 168, 180, 189, 210, 216, 240, 252, 270, 280, 315, 324, 336, 360, 378, 405, 420, 432, 504, 540, 560, 567, 630, 648, 720, 756, 810, 840, 945, 1008, 1080, 1134, 1260, 1296, 1512, 1620, 1680, 1890, 2160, 2268, 2520, 2835, 3024, 3240, 3780, 4536, 5040, 5670, 6480, 7560, 9072, 11340, 15120, 22680, 45360

Answer 2

Actually, Scott R's answer is *not* correct:

"45360" has only 4 unique factors: 2,3,5,7

and "100" has only 2 unique factors: 2 and 5

But you can see that the factors of these numbers are prime numbers! So, all you have to do is multiply the first 99 prime numbers together to find your answer! (Normally it would be the first 100 prime factors, but you specifically wanted "1" to be a factor) The first 99 prime numbers are:

2. 3. 5. 7. 11. 13. 17. 19. 23. 29. 31. 37. 41. 43. 47. 53. 59. 61. 67. 71. 73. 79. 83. 89. 97. 101. 103. 107. 109. 113. 127. 131. 137. 139. 149. 151. 157. 163. 167. 173. 179. 181. 191. 193. 197. 199. 211. 223. 227. 229. 233. 239. 241. 251. 257. 263. 269. 271. 277. 281. 283. 293. 307. 311. 313. 317. 331. 337. 347. 349. 353. 359. 367. 373. 379. 383. 389. 397. 401. 409. 419. 421. 431. 433. 439. 443. 449. 457. 461. 463. 467. 479. 487. 491. 499. 503. 509. 521. 523.


When you multiply these numbers together, you get:
8.70966876137927e+216

Most people do not include 1 as a factor, so the smallest integer with exactly 100 factors is a number which has the first 100 prime numbers as its factors!

The 100th prime number is: 541, so if you multiply the first 100 prime numbers together, you get: 4.71193079990619e+219.

Good question!

Answer 3

1^100= 1

or the center number on step 101 of Pascals Triangle
TRIP ON THIS
Step 1 = 1
Step 2 = 1 1
Step 3 = 1 2 1 starting step 3 and only for the odd numbered steps, subtract the step number -1 to get the number of factors for the middle number 2 = 2*1 = 3-1
Step 4 = 1 3 3 1
Step 5 = 1 4 6 4 1 [6 = 1*6, 2*3 ( 4 factors ) = 5-1]
Step 6 = 1 5 10 10 5 1
Step 7 = 1 6 15 20 15 6 1 [20 = 1*20, 10*2, 4*5 (6 factors) = 7-1]
Step 8 = 1 7 21 35 35 21 7 1
Step 9 = 1 8 28 56 70 56 28 8 1 [70= 1*70, 7*2*5, 2*35, 14*5, 7*10 (8 factors 1,2,5,7,10,14,35,70) = 9-1]
or in case repeat 1's and Pascal don't count.

1 + 2^98 = x
316912650057057350374175801345 =
1, 2 (98 times) and 316912650057057350374175801345

Answer 4

It suspect it's the product of the first 100 prime numbers.