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Type | Label | Description |
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Statement | ||
Theorem | mulmoddvds 11901 | If an integer is divisible by a positive integer, the product of this integer with another integer modulo the positive integer is 0. (Contributed by Alexander van der Vekens, 30-Aug-2018.) |
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Theorem | 3dvdsdec 11902 |
A decimal number is divisible by three iff the sum of its two
"digits"
is divisible by three. The term "digits" in its narrow sense
is only
correct if ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | 3dvds2dec 11903 |
A decimal number is divisible by three iff the sum of its three
"digits"
is divisible by three. The term "digits" in its narrow sense
is only
correct if ![]() ![]() ![]() ![]() ![]() ![]() |
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The set | ||
Theorem | evenelz 11904 | An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 11831. (Contributed by AV, 22-Jun-2021.) |
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Theorem | zeo3 11905 | An integer is even or odd. (Contributed by AV, 17-Jun-2021.) |
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Theorem | zeoxor 11906 | An integer is even or odd but not both. (Contributed by Jim Kingdon, 10-Nov-2021.) |
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Theorem | zeo4 11907 | An integer is even or odd but not both. (Contributed by AV, 17-Jun-2021.) |
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Theorem | zeneo 11908 | No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. This variant of zneo 9384 follows immediately from the fact that a contradiction implies anything, see pm2.21i 647. (Contributed by AV, 22-Jun-2021.) |
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Theorem | odd2np1lem 11909* | Lemma for odd2np1 11910. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
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Theorem | odd2np1 11910* | An integer is odd iff it is one plus twice another integer. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
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Theorem | even2n 11911* | An integer is even iff it is twice another integer. (Contributed by AV, 25-Jun-2020.) |
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Theorem | oddm1even 11912 | An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) |
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Theorem | oddp1even 11913 | An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) |
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Theorem | oexpneg 11914 | The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.) |
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Theorem | mod2eq0even 11915 | An integer is 0 modulo 2 iff it is even (i.e. divisible by 2), see example 2 in [ApostolNT] p. 107. (Contributed by AV, 21-Jul-2021.) |
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Theorem | mod2eq1n2dvds 11916 | An integer is 1 modulo 2 iff it is odd (i.e. not divisible by 2), see example 3 in [ApostolNT] p. 107. (Contributed by AV, 24-May-2020.) |
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Theorem | oddnn02np1 11917* | A nonnegative integer is odd iff it is one plus twice another nonnegative integer. (Contributed by AV, 19-Jun-2021.) |
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Theorem | oddge22np1 11918* | An integer greater than one is odd iff it is one plus twice a positive integer. (Contributed by AV, 16-Aug-2021.) |
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Theorem | evennn02n 11919* | A nonnegative integer is even iff it is twice another nonnegative integer. (Contributed by AV, 12-Aug-2021.) |
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Theorem | evennn2n 11920* | A positive integer is even iff it is twice another positive integer. (Contributed by AV, 12-Aug-2021.) |
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Theorem | 2tp1odd 11921 | A number which is twice an integer increased by 1 is odd. (Contributed by AV, 16-Jul-2021.) |
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Theorem | mulsucdiv2z 11922 | An integer multiplied with its successor divided by 2 yields an integer, i.e. an integer multiplied with its successor is even. (Contributed by AV, 19-Jul-2021.) |
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Theorem | sqoddm1div8z 11923 | A squared odd number minus 1 divided by 8 is an integer. (Contributed by AV, 19-Jul-2021.) |
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Theorem | 2teven 11924 | A number which is twice an integer is even. (Contributed by AV, 16-Jul-2021.) |
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Theorem | zeo5 11925 | An integer is either even or odd, version of zeo3 11905 avoiding the negation of the representation of an odd number. (Proposed by BJ, 21-Jun-2021.) (Contributed by AV, 26-Jun-2020.) |
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Theorem | evend2 11926 | An integer is even iff its quotient with 2 is an integer. This is a representation of even numbers without using the divides relation, see zeo 9388 and zeo2 9389. (Contributed by AV, 22-Jun-2021.) |
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Theorem | oddp1d2 11927 | An integer is odd iff its successor divided by 2 is an integer. This is a representation of odd numbers without using the divides relation, see zeo 9388 and zeo2 9389. (Contributed by AV, 22-Jun-2021.) |
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Theorem | zob 11928 | Alternate characterizations of an odd number. (Contributed by AV, 7-Jun-2020.) |
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Theorem | oddm1d2 11929 | An integer is odd iff its predecessor divided by 2 is an integer. This is another representation of odd numbers without using the divides relation. (Contributed by AV, 18-Jun-2021.) (Proof shortened by AV, 22-Jun-2021.) |
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Theorem | ltoddhalfle 11930 | An integer is less than half of an odd number iff it is less than or equal to the half of the predecessor of the odd number (which is an even number). (Contributed by AV, 29-Jun-2021.) |
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Theorem | halfleoddlt 11931 | An integer is greater than half of an odd number iff it is greater than or equal to the half of the odd number. (Contributed by AV, 1-Jul-2021.) |
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Theorem | opoe 11932 | The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
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Theorem | omoe 11933 | The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
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Theorem | opeo 11934 | The sum of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
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Theorem | omeo 11935 | The difference of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
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Theorem | m1expe 11936 | Exponentiation of -1 by an even power. Variant of m1expeven 10598. (Contributed by AV, 25-Jun-2021.) |
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Theorem | m1expo 11937 | Exponentiation of -1 by an odd power. (Contributed by AV, 26-Jun-2021.) |
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Theorem | m1exp1 11938 | Exponentiation of negative one is one iff the exponent is even. (Contributed by AV, 20-Jun-2021.) |
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Theorem | nn0enne 11939 | A positive integer is an even nonnegative integer iff it is an even positive integer. (Contributed by AV, 30-May-2020.) |
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Theorem | nn0ehalf 11940 | The half of an even nonnegative integer is a nonnegative integer. (Contributed by AV, 22-Jun-2020.) (Revised by AV, 28-Jun-2021.) |
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Theorem | nnehalf 11941 | The half of an even positive integer is a positive integer. (Contributed by AV, 28-Jun-2021.) |
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Theorem | nn0o1gt2 11942 | An odd nonnegative integer is either 1 or greater than 2. (Contributed by AV, 2-Jun-2020.) |
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Theorem | nno 11943 | An alternate characterization of an odd integer greater than 1. (Contributed by AV, 2-Jun-2020.) |
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Theorem | nn0o 11944 | An alternate characterization of an odd nonnegative integer. (Contributed by AV, 28-May-2020.) (Proof shortened by AV, 2-Jun-2020.) |
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Theorem | nn0ob 11945 | Alternate characterizations of an odd nonnegative integer. (Contributed by AV, 4-Jun-2020.) |
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Theorem | nn0oddm1d2 11946 | A positive integer is odd iff its predecessor divided by 2 is a positive integer. (Contributed by AV, 28-Jun-2021.) |
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Theorem | nnoddm1d2 11947 | A positive integer is odd iff its successor divided by 2 is a positive integer. (Contributed by AV, 28-Jun-2021.) |
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Theorem | z0even 11948 | 0 is even. (Contributed by AV, 11-Feb-2020.) (Revised by AV, 23-Jun-2021.) |
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Theorem | n2dvds1 11949 | 2 does not divide 1 (common case). That means 1 is odd. (Contributed by David A. Wheeler, 8-Dec-2018.) |
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Theorem | n2dvdsm1 11950 | 2 does not divide -1. That means -1 is odd. (Contributed by AV, 15-Aug-2021.) |
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Theorem | z2even 11951 | 2 is even. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 23-Jun-2021.) |
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Theorem | n2dvds3 11952 | 2 does not divide 3, i.e. 3 is an odd number. (Contributed by AV, 28-Feb-2021.) |
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Theorem | z4even 11953 | 4 is an even number. (Contributed by AV, 23-Jul-2020.) (Revised by AV, 4-Jul-2021.) |
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Theorem | 4dvdseven 11954 | An integer which is divisible by 4 is an even integer. (Contributed by AV, 4-Jul-2021.) |
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Theorem | divalglemnn 11955* | Lemma for divalg 11961. Existence for a positive denominator. (Contributed by Jim Kingdon, 30-Nov-2021.) |
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Theorem | divalglemqt 11956 |
Lemma for divalg 11961. The ![]() ![]() ![]() |
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Theorem | divalglemnqt 11957 |
Lemma for divalg 11961. The ![]() ![]() ![]() |
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Theorem | divalglemeunn 11958* | Lemma for divalg 11961. Uniqueness for a positive denominator. (Contributed by Jim Kingdon, 4-Dec-2021.) |
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Theorem | divalglemex 11959* | Lemma for divalg 11961. The quotient and remainder exist. (Contributed by Jim Kingdon, 30-Nov-2021.) |
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Theorem | divalglemeuneg 11960* | Lemma for divalg 11961. Uniqueness for a negative denominator. (Contributed by Jim Kingdon, 4-Dec-2021.) |
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Theorem | divalg 11961* |
The division algorithm (theorem). Dividing an integer ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | divalgb 11962* |
Express the division algorithm as stated in divalg 11961 in terms of
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Theorem | divalg2 11963* | The division algorithm (theorem) for a positive divisor. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | divalgmod 11964 |
The result of the ![]() ![]() ![]() |
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Theorem | divalgmodcl 11965 |
The result of the ![]() ![]() |
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Theorem | modremain 11966* | The result of the modulo operation is the remainder of the division algorithm. (Contributed by AV, 19-Aug-2021.) |
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Theorem | ndvdssub 11967 |
Corollary of the division algorithm. If an integer ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | ndvdsadd 11968 |
Corollary of the division algorithm. If an integer ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | ndvdsp1 11969 |
Special case of ndvdsadd 11968. If an integer ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | ndvdsi 11970 | A quick test for non-divisibility. (Contributed by Mario Carneiro, 18-Feb-2014.) |
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Theorem | flodddiv4 11971 | The floor of an odd integer divided by 4. (Contributed by AV, 17-Jun-2021.) |
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Theorem | fldivndvdslt 11972 | The floor of an integer divided by a nonzero integer not dividing the first integer is less than the integer divided by the positive integer. (Contributed by AV, 4-Jul-2021.) |
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Theorem | flodddiv4lt 11973 | The floor of an odd number divided by 4 is less than the odd number divided by 4. (Contributed by AV, 4-Jul-2021.) |
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Theorem | flodddiv4t2lthalf 11974 | The floor of an odd number divided by 4, multiplied by 2 is less than the half of the odd number. (Contributed by AV, 4-Jul-2021.) |
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Syntax | cgcd 11975 | Extend the definition of a class to include the greatest common divisor operator. |
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Definition | df-gcd 11976* |
Define the ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | gcdmndc 11977 |
Decidablity lemma used in various proofs related to ![]() |
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Theorem | zsupcllemstep 11978* | Lemma for zsupcl 11980. Induction step. (Contributed by Jim Kingdon, 7-Dec-2021.) |
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Theorem | zsupcllemex 11979* | Lemma for zsupcl 11980. Existence of the supremum. (Contributed by Jim Kingdon, 7-Dec-2021.) |
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Theorem | zsupcl 11980* |
Closure of supremum for decidable integer properties. The property
which defines the set we are taking the supremum of must (a) be true at
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Theorem | zssinfcl 11981* | The infimum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 16-Jan-2022.) |
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Theorem | infssuzex 11982* | Existence of the infimum of a subset of an upper set of integers. (Contributed by Jim Kingdon, 13-Jan-2022.) |
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Theorem | infssuzledc 11983* | The infimum of a subset of an upper set of integers is less than or equal to all members of the subset. (Contributed by Jim Kingdon, 13-Jan-2022.) |
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Theorem | infssuzcldc 11984* | The infimum of a subset of an upper set of integers belongs to the subset. (Contributed by Jim Kingdon, 20-Jan-2022.) |
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Theorem | suprzubdc 11985* | The supremum of a bounded-above decidable set of integers is greater than any member of the set. (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 5-Oct-2024.) |
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Theorem | nninfdcex 11986* | A decidable set of natural numbers has an infimum. (Contributed by Jim Kingdon, 28-Sep-2024.) |
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Theorem | zsupssdc 11987* | An inhabited decidable bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-suploc 7962.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 5-Oct-2024.) |
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Theorem | suprzcl2dc 11988* | The supremum of a bounded-above decidable set of integers is a member of the set. (This theorem avoids ax-pre-suploc 7962.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 6-Oct-2024.) |
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Theorem | dvdsbnd 11989* | There is an upper bound to the divisors of a nonzero integer. (Contributed by Jim Kingdon, 11-Dec-2021.) |
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Theorem | gcdsupex 11990* |
Existence of the supremum used in defining ![]() |
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Theorem | gcdsupcl 11991* |
Closure of the supremum used in defining ![]() |
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Theorem | gcdval 11992* |
The value of the ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | gcd0val 11993 |
The value, by convention, of the ![]() |
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Theorem | gcdn0val 11994* |
The value of the ![]() |
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Theorem | gcdn0cl 11995 |
Closure of the ![]() |
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Theorem | gcddvds 11996 | The gcd of two integers divides each of them. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdslegcd 11997 |
An integer which divides both operands of the ![]() |
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Theorem | nndvdslegcd 11998 |
A positive integer which divides both positive operands of the ![]() |
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Theorem | gcdcl 11999 |
Closure of the ![]() |
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Theorem | gcdnncl 12000 |
Closure of the ![]() |
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