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batch-20260926-factorial-foundations · 2026-09-26 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Compute n! for the natural number n=0; give the answer as a decimal integer string.
1
0!=1, computed exactly with arbitrary-precision integer arithmetic.
Reviewed step-by-step derivation
Factorial · qf-factorial-exact-evaluation · candidate
Compute n! for the natural number n=1; give the answer as a decimal integer string.
1
1!=1, computed exactly with arbitrary-precision integer arithmetic.
Reviewed step-by-step derivation
Factorial · qf-factorial-exact-evaluation · candidate
Compute n! for the natural number n=2; give the answer as a decimal integer string.
2
2!=2, computed exactly with arbitrary-precision integer arithmetic.
Reviewed step-by-step derivation
Factorial · qf-factorial-exact-evaluation · candidate
Compute n! for the natural number n=3; give the answer as a decimal integer string.
6
3!=6, computed exactly with arbitrary-precision integer arithmetic.
Reviewed step-by-step derivation
Factorial · qf-factorial-exact-evaluation · candidate
Compute n! for the natural number n=4; give the answer as a decimal integer string.
24
4!=24, computed exactly with arbitrary-precision integer arithmetic.
Reviewed step-by-step derivation
Factorial · qf-factorial-exact-evaluation · candidate
Compute n! for the natural number n=5; give the answer as a decimal integer string.
120
5!=120, computed exactly with arbitrary-precision integer arithmetic.
Reviewed step-by-step derivation
Factorial · qf-factorial-exact-evaluation · candidate
Compute n! for the natural number n=6; give the answer as a decimal integer string.
720
6!=720, computed exactly with arbitrary-precision integer arithmetic.
Reviewed step-by-step derivation
Factorial · qf-factorial-exact-evaluation · candidate
Compute n! for the natural number n=7; give the answer as a decimal integer string.
5040
7!=5040, computed exactly with arbitrary-precision integer arithmetic.
Reviewed step-by-step derivation
Factorial · qf-factorial-exact-evaluation · candidate
Compute n! for the natural number n=8; give the answer as a decimal integer string.
40320
8!=40320, computed exactly with arbitrary-precision integer arithmetic.
Reviewed step-by-step derivation
Factorial · qf-factorial-exact-evaluation · candidate
Compute n! for the natural number n=9; give the answer as a decimal integer string.
362880
9!=362880, computed exactly with arbitrary-precision integer arithmetic.
Reviewed step-by-step derivation
Factorial · qf-factorial-exact-evaluation · candidate
Compute n! for the natural number n=10; give the answer as a decimal integer string.
3628800
10!=3628800, computed exactly with arbitrary-precision integer arithmetic.
Reviewed step-by-step derivation
Factorial · qf-factorial-exact-evaluation · candidate
Compute n! for the natural number n=11; give the answer as a decimal integer string.
39916800
11!=39916800, computed exactly with arbitrary-precision integer arithmetic.
Reviewed step-by-step derivation
Factorial · qf-factorial-exact-evaluation · candidate
Compute n! for the natural number n=12; give the answer as a decimal integer string.
479001600
12!=479001600, computed exactly with arbitrary-precision integer arithmetic.
Reviewed step-by-step derivation
Factorial · qf-factorial-exact-evaluation · candidate
Compute n! for the natural number n=13; give the answer as a decimal integer string.
6227020800
13!=6227020800, computed exactly with arbitrary-precision integer arithmetic.
Reviewed step-by-step derivation
Factorial · qf-factorial-exact-evaluation · candidate
Compute n! for the natural number n=14; give the answer as a decimal integer string.
87178291200
14!=87178291200, computed exactly with arbitrary-precision integer arithmetic.
Reviewed step-by-step derivation
Factorial · qf-factorial-exact-evaluation · candidate
Compute n! for the natural number n=15; give the answer as a decimal integer string.
1307674368000
15!=1307674368000, computed exactly with arbitrary-precision integer arithmetic.
Reviewed step-by-step derivation
Factorial · qf-factorial-exact-evaluation · candidate
Compute n! for the natural number n=16; give the answer as a decimal integer string.
20922789888000
16!=20922789888000, computed exactly with arbitrary-precision integer arithmetic.
Reviewed step-by-step derivation
Factorial · qf-factorial-exact-evaluation · candidate
Compute n! for the natural number n=17; give the answer as a decimal integer string.
355687428096000
17!=355687428096000, computed exactly with arbitrary-precision integer arithmetic.
Reviewed step-by-step derivation
Factorial · qf-factorial-exact-evaluation · candidate
Compute n! for the natural number n=18; give the answer as a decimal integer string.
6402373705728000
18!=6402373705728000, computed exactly with arbitrary-precision integer arithmetic.
Reviewed step-by-step derivation
Factorial · qf-factorial-exact-evaluation · candidate
Compute n! for the natural number n=19; give the answer as a decimal integer string.
121645100408832000
19!=121645100408832000, computed exactly with arbitrary-precision integer arithmetic.
Reviewed step-by-step derivation
Factorial · qf-factorial-exact-evaluation · candidate
Compute n! for the natural number n=20; give the answer as a decimal integer string.
2432902008176640000
20!=2432902008176640000, computed exactly with arbitrary-precision integer arithmetic.
Reviewed step-by-step derivation
Factorial · qf-factorial-exact-evaluation · candidate
Compute n! for the natural number n=21; give the answer as a decimal integer string.
51090942171709440000
21!=51090942171709440000, computed exactly with arbitrary-precision integer arithmetic.
Reviewed step-by-step derivation
Factorial · qf-factorial-exact-evaluation · candidate
Compute n! for the natural number n=22; give the answer as a decimal integer string.
1124000727777607680000
22!=1124000727777607680000, computed exactly with arbitrary-precision integer arithmetic.
Reviewed step-by-step derivation
Factorial · qf-factorial-exact-evaluation · candidate
Compute n! for the natural number n=23; give the answer as a decimal integer string.
25852016738884976640000
23!=25852016738884976640000, computed exactly with arbitrary-precision integer arithmetic.
Reviewed step-by-step derivation
Factorial · qf-factorial-exact-evaluation · candidate
Compute n! for the natural number n=24; give the answer as a decimal integer string.
620448401733239439360000
24!=620448401733239439360000, computed exactly with arbitrary-precision integer arithmetic.
Reviewed step-by-step derivation
Factorial · qf-factorial-exact-evaluation · candidate
Given 0!=1, use (n+1)!=(n+1)×n! to find 1!.
1
1×1=1, so 1!=1.
Reviewed step-by-step derivation
Factorial · qf-factorial-successor-recurrence · candidate
Given 1!=1, use (n+1)!=(n+1)×n! to find 2!.
2
2×1=2, so 2!=2.
Reviewed step-by-step derivation
Factorial · qf-factorial-successor-recurrence · candidate
Given 2!=2, use (n+1)!=(n+1)×n! to find 3!.
6
3×2=6, so 3!=6.
Reviewed step-by-step derivation
Factorial · qf-factorial-successor-recurrence · candidate
Given 3!=6, use (n+1)!=(n+1)×n! to find 4!.
24
4×6=24, so 4!=24.
Reviewed step-by-step derivation
Factorial · qf-factorial-successor-recurrence · candidate
Given 4!=24, use (n+1)!=(n+1)×n! to find 5!.
120
5×24=120, so 5!=120.
Reviewed step-by-step derivation
Factorial · qf-factorial-successor-recurrence · candidate
Given 5!=120, use (n+1)!=(n+1)×n! to find 6!.
720
6×120=720, so 6!=720.
Reviewed step-by-step derivation
Factorial · qf-factorial-successor-recurrence · candidate
Given 6!=720, use (n+1)!=(n+1)×n! to find 7!.
5040
7×720=5040, so 7!=5040.
Reviewed step-by-step derivation
Factorial · qf-factorial-successor-recurrence · candidate
Given 7!=5040, use (n+1)!=(n+1)×n! to find 8!.
40320
8×5040=40320, so 8!=40320.
Reviewed step-by-step derivation
Factorial · qf-factorial-successor-recurrence · candidate
Given 8!=40320, use (n+1)!=(n+1)×n! to find 9!.
362880
9×40320=362880, so 9!=362880.
Reviewed step-by-step derivation
Factorial · qf-factorial-successor-recurrence · candidate
Given 9!=362880, use (n+1)!=(n+1)×n! to find 10!.
3628800
10×362880=3628800, so 10!=3628800.
Reviewed step-by-step derivation
Factorial · qf-factorial-successor-recurrence · candidate
Given 10!=3628800, use (n+1)!=(n+1)×n! to find 11!.
39916800
11×3628800=39916800, so 11!=39916800.
Reviewed step-by-step derivation
Factorial · qf-factorial-successor-recurrence · candidate
Given 11!=39916800, use (n+1)!=(n+1)×n! to find 12!.
479001600
12×39916800=479001600, so 12!=479001600.
Reviewed step-by-step derivation
Factorial · qf-factorial-successor-recurrence · candidate
Given 12!=479001600, use (n+1)!=(n+1)×n! to find 13!.
6227020800
13×479001600=6227020800, so 13!=6227020800.
Reviewed step-by-step derivation
Factorial · qf-factorial-successor-recurrence · candidate
Given 13!=6227020800, use (n+1)!=(n+1)×n! to find 14!.
87178291200
14×6227020800=87178291200, so 14!=87178291200.
Reviewed step-by-step derivation
Factorial · qf-factorial-successor-recurrence · candidate
Given 14!=87178291200, use (n+1)!=(n+1)×n! to find 15!.
1307674368000
15×87178291200=1307674368000, so 15!=1307674368000.
Reviewed step-by-step derivation
Factorial · qf-factorial-successor-recurrence · candidate
Given 15!=1307674368000, use (n+1)!=(n+1)×n! to find 16!.
20922789888000
16×1307674368000=20922789888000, so 16!=20922789888000.
Reviewed step-by-step derivation
Factorial · qf-factorial-successor-recurrence · candidate
Given 16!=20922789888000, use (n+1)!=(n+1)×n! to find 17!.
355687428096000
17×20922789888000=355687428096000, so 17!=355687428096000.
Reviewed step-by-step derivation
Factorial · qf-factorial-successor-recurrence · candidate
Given 17!=355687428096000, use (n+1)!=(n+1)×n! to find 18!.
6402373705728000
18×355687428096000=6402373705728000, so 18!=6402373705728000.
Reviewed step-by-step derivation
Factorial · qf-factorial-successor-recurrence · candidate
Given 18!=6402373705728000, use (n+1)!=(n+1)×n! to find 19!.
121645100408832000
19×6402373705728000=121645100408832000, so 19!=121645100408832000.
Reviewed step-by-step derivation
Factorial · qf-factorial-successor-recurrence · candidate
Given 19!=121645100408832000, use (n+1)!=(n+1)×n! to find 20!.
2432902008176640000
20×121645100408832000=2432902008176640000, so 20!=2432902008176640000.
Reviewed step-by-step derivation
Factorial · qf-factorial-successor-recurrence · candidate
Given 20!=2432902008176640000, use (n+1)!=(n+1)×n! to find 21!.
51090942171709440000
21×2432902008176640000=51090942171709440000, so 21!=51090942171709440000.
Reviewed step-by-step derivation
Factorial · qf-factorial-successor-recurrence · candidate
Given 21!=51090942171709440000, use (n+1)!=(n+1)×n! to find 22!.
1124000727777607680000
22×51090942171709440000=1124000727777607680000, so 22!=1124000727777607680000.
Reviewed step-by-step derivation
Factorial · qf-factorial-successor-recurrence · candidate
Given 22!=1124000727777607680000, use (n+1)!=(n+1)×n! to find 23!.
25852016738884976640000
23×1124000727777607680000=25852016738884976640000, so 23!=25852016738884976640000.
Reviewed step-by-step derivation
Factorial · qf-factorial-successor-recurrence · candidate
Given 23!=25852016738884976640000, use (n+1)!=(n+1)×n! to find 24!.
620448401733239439360000
24×25852016738884976640000=620448401733239439360000, so 24!=620448401733239439360000.
Reviewed step-by-step derivation
Factorial · qf-factorial-successor-recurrence · candidate
Given 24!=620448401733239439360000, use (n+1)!=(n+1)×n! to find 25!.
15511210043330985984000000
25×620448401733239439360000=15511210043330985984000000, so 25!=15511210043330985984000000.
Reviewed step-by-step derivation
Factorial · qf-factorial-successor-recurrence · candidate
Compute 1!/0!; the denominator is explicitly positive.
1
1/1=1, using 1!=1×0!.
Reviewed step-by-step derivation
Factorial · qf-factorial-consecutive-quotient · candidate
Compute 2!/1!; the denominator is explicitly positive.
2
2/1=2, using 2!=2×1!.
Reviewed step-by-step derivation
Factorial · qf-factorial-consecutive-quotient · candidate
Compute 3!/2!; the denominator is explicitly positive.
3
6/2=3, using 3!=3×2!.
Reviewed step-by-step derivation
Factorial · qf-factorial-consecutive-quotient · candidate
Compute 4!/3!; the denominator is explicitly positive.
4
24/6=4, using 4!=4×3!.
Reviewed step-by-step derivation
Factorial · qf-factorial-consecutive-quotient · candidate
Compute 5!/4!; the denominator is explicitly positive.
5
120/24=5, using 5!=5×4!.
Reviewed step-by-step derivation
Factorial · qf-factorial-consecutive-quotient · candidate
Compute 6!/5!; the denominator is explicitly positive.
6
720/120=6, using 6!=6×5!.
Reviewed step-by-step derivation
Factorial · qf-factorial-consecutive-quotient · candidate
Compute 7!/6!; the denominator is explicitly positive.
7
5040/720=7, using 7!=7×6!.
Reviewed step-by-step derivation
Factorial · qf-factorial-consecutive-quotient · candidate
Compute 8!/7!; the denominator is explicitly positive.
8
40320/5040=8, using 8!=8×7!.
Reviewed step-by-step derivation
Factorial · qf-factorial-consecutive-quotient · candidate
Compute 9!/8!; the denominator is explicitly positive.
9
362880/40320=9, using 9!=9×8!.
Reviewed step-by-step derivation
Factorial · qf-factorial-consecutive-quotient · candidate
Compute 10!/9!; the denominator is explicitly positive.
10
3628800/362880=10, using 10!=10×9!.
Reviewed step-by-step derivation
Factorial · qf-factorial-consecutive-quotient · candidate
Compute 11!/10!; the denominator is explicitly positive.
11
39916800/3628800=11, using 11!=11×10!.
Reviewed step-by-step derivation
Factorial · qf-factorial-consecutive-quotient · candidate
Compute 12!/11!; the denominator is explicitly positive.
12
479001600/39916800=12, using 12!=12×11!.
Reviewed step-by-step derivation
Factorial · qf-factorial-consecutive-quotient · candidate
Compute 13!/12!; the denominator is explicitly positive.
13
6227020800/479001600=13, using 13!=13×12!.
Reviewed step-by-step derivation
Factorial · qf-factorial-consecutive-quotient · candidate
Compute 14!/13!; the denominator is explicitly positive.
14
87178291200/6227020800=14, using 14!=14×13!.
Reviewed step-by-step derivation
Factorial · qf-factorial-consecutive-quotient · candidate
Compute 15!/14!; the denominator is explicitly positive.
15
1307674368000/87178291200=15, using 15!=15×14!.
Reviewed step-by-step derivation
Factorial · qf-factorial-consecutive-quotient · candidate
Compute 16!/15!; the denominator is explicitly positive.
16
20922789888000/1307674368000=16, using 16!=16×15!.
Reviewed step-by-step derivation
Factorial · qf-factorial-consecutive-quotient · candidate
Compute 17!/16!; the denominator is explicitly positive.
17
355687428096000/20922789888000=17, using 17!=17×16!.
Reviewed step-by-step derivation
Factorial · qf-factorial-consecutive-quotient · candidate
Compute 18!/17!; the denominator is explicitly positive.
18
6402373705728000/355687428096000=18, using 18!=18×17!.
Reviewed step-by-step derivation
Factorial · qf-factorial-consecutive-quotient · candidate
Compute 19!/18!; the denominator is explicitly positive.
19
121645100408832000/6402373705728000=19, using 19!=19×18!.
Reviewed step-by-step derivation
Factorial · qf-factorial-consecutive-quotient · candidate
Compute 20!/19!; the denominator is explicitly positive.
20
2432902008176640000/121645100408832000=20, using 20!=20×19!.
Reviewed step-by-step derivation
Factorial · qf-factorial-consecutive-quotient · candidate
Compute 21!/20!; the denominator is explicitly positive.
21
51090942171709440000/2432902008176640000=21, using 21!=21×20!.
Reviewed step-by-step derivation
Factorial · qf-factorial-consecutive-quotient · candidate
Compute 22!/21!; the denominator is explicitly positive.
22
1124000727777607680000/51090942171709440000=22, using 22!=22×21!.
Reviewed step-by-step derivation
Factorial · qf-factorial-consecutive-quotient · candidate
Compute 23!/22!; the denominator is explicitly positive.
23
25852016738884976640000/1124000727777607680000=23, using 23!=23×22!.
Reviewed step-by-step derivation
Factorial · qf-factorial-consecutive-quotient · candidate
Compute 24!/23!; the denominator is explicitly positive.
24
620448401733239439360000/25852016738884976640000=24, using 24!=24×23!.
Reviewed step-by-step derivation
Factorial · qf-factorial-consecutive-quotient · candidate
Compute 25!/24!; the denominator is explicitly positive.
25
15511210043330985984000000/620448401733239439360000=25, using 25!=25×24!.
Reviewed step-by-step derivation
Factorial · qf-factorial-consecutive-quotient · candidate
Fill the integer □ in 1!=□×0!, where 0!=1.
1
1= 1×1, so □=1.
Reviewed step-by-step derivation
Factorial · qf-factorial-missing-successor-factor · candidate
Fill the integer □ in 2!=□×1!, where 1!=1.
2
2= 2×1, so □=2.
Reviewed step-by-step derivation
Factorial · qf-factorial-missing-successor-factor · candidate
Fill the integer □ in 3!=□×2!, where 2!=2.
3
6= 3×2, so □=3.
Reviewed step-by-step derivation
Factorial · qf-factorial-missing-successor-factor · candidate
Fill the integer □ in 4!=□×3!, where 3!=6.
4
24= 4×6, so □=4.
Reviewed step-by-step derivation
Factorial · qf-factorial-missing-successor-factor · candidate
Fill the integer □ in 5!=□×4!, where 4!=24.
5
120= 5×24, so □=5.
Reviewed step-by-step derivation
Factorial · qf-factorial-missing-successor-factor · candidate
Fill the integer □ in 6!=□×5!, where 5!=120.
6
720= 6×120, so □=6.
Reviewed step-by-step derivation
Factorial · qf-factorial-missing-successor-factor · candidate
Fill the integer □ in 7!=□×6!, where 6!=720.
7
5040= 7×720, so □=7.
Reviewed step-by-step derivation
Factorial · qf-factorial-missing-successor-factor · candidate
Fill the integer □ in 8!=□×7!, where 7!=5040.
8
40320= 8×5040, so □=8.
Reviewed step-by-step derivation
Factorial · qf-factorial-missing-successor-factor · candidate
Fill the integer □ in 9!=□×8!, where 8!=40320.
9
362880= 9×40320, so □=9.
Reviewed step-by-step derivation
Factorial · qf-factorial-missing-successor-factor · candidate
Fill the integer □ in 10!=□×9!, where 9!=362880.
10
3628800= 10×362880, so □=10.
Reviewed step-by-step derivation
Factorial · qf-factorial-missing-successor-factor · candidate
Fill the integer □ in 11!=□×10!, where 10!=3628800.
11
39916800= 11×3628800, so □=11.
Reviewed step-by-step derivation
Factorial · qf-factorial-missing-successor-factor · candidate
Fill the integer □ in 12!=□×11!, where 11!=39916800.
12
479001600= 12×39916800, so □=12.
Reviewed step-by-step derivation
Factorial · qf-factorial-missing-successor-factor · candidate
Fill the integer □ in 13!=□×12!, where 12!=479001600.
13
6227020800= 13×479001600, so □=13.
Reviewed step-by-step derivation
Factorial · qf-factorial-missing-successor-factor · candidate
Fill the integer □ in 14!=□×13!, where 13!=6227020800.
14
87178291200= 14×6227020800, so □=14.
Reviewed step-by-step derivation
Factorial · qf-factorial-missing-successor-factor · candidate
Fill the integer □ in 15!=□×14!, where 14!=87178291200.
15
1307674368000= 15×87178291200, so □=15.
Reviewed step-by-step derivation
Factorial · qf-factorial-missing-successor-factor · candidate
Fill the integer □ in 16!=□×15!, where 15!=1307674368000.
16
20922789888000= 16×1307674368000, so □=16.
Reviewed step-by-step derivation
Factorial · qf-factorial-missing-successor-factor · candidate
Fill the integer □ in 17!=□×16!, where 16!=20922789888000.
17
355687428096000= 17×20922789888000, so □=17.
Reviewed step-by-step derivation
Factorial · qf-factorial-missing-successor-factor · candidate
Fill the integer □ in 18!=□×17!, where 17!=355687428096000.
18
6402373705728000= 18×355687428096000, so □=18.
Reviewed step-by-step derivation
Factorial · qf-factorial-missing-successor-factor · candidate
Fill the integer □ in 19!=□×18!, where 18!=6402373705728000.
19
121645100408832000= 19×6402373705728000, so □=19.
Reviewed step-by-step derivation
Factorial · qf-factorial-missing-successor-factor · candidate
Fill the integer □ in 20!=□×19!, where 19!=121645100408832000.
20
2432902008176640000= 20×121645100408832000, so □=20.
Reviewed step-by-step derivation
Factorial · qf-factorial-missing-successor-factor · candidate
Fill the integer □ in 21!=□×20!, where 20!=2432902008176640000.
21
51090942171709440000= 21×2432902008176640000, so □=21.
Reviewed step-by-step derivation
Factorial · qf-factorial-missing-successor-factor · candidate
Fill the integer □ in 22!=□×21!, where 21!=51090942171709440000.
22
1124000727777607680000= 22×51090942171709440000, so □=22.
Reviewed step-by-step derivation
Factorial · qf-factorial-missing-successor-factor · candidate
Fill the integer □ in 23!=□×22!, where 22!=1124000727777607680000.
23
25852016738884976640000= 23×1124000727777607680000, so □=23.
Reviewed step-by-step derivation
Factorial · qf-factorial-missing-successor-factor · candidate
Fill the integer □ in 24!=□×23!, where 23!=25852016738884976640000.
24
620448401733239439360000= 24×25852016738884976640000, so □=24.
Reviewed step-by-step derivation
Factorial · qf-factorial-missing-successor-factor · candidate
Fill the integer □ in 25!=□×24!, where 24!=620448401733239439360000.
25
15511210043330985984000000= 25×620448401733239439360000, so □=25.
Reviewed step-by-step derivation
Factorial · qf-factorial-missing-successor-factor · candidate
Canonical question JSONL · English JSONL · Validation report