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Type | Label | Description |
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Statement | ||
Theorem | cos01bnd 11501 | Bounds on the cosine of a positive real number less than or equal to 1. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.) |
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Theorem | cos1bnd 11502 | Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.) |
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Theorem | cos2bnd 11503 | Bounds on the cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.) |
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Theorem | sin01gt0 11504 | The sine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Wolf Lammen, 25-Sep-2020.) |
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Theorem | cos01gt0 11505 | The cosine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.) |
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Theorem | sin02gt0 11506 | The sine of a positive real number less than or equal to 2 is positive. (Contributed by Paul Chapman, 19-Jan-2008.) |
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Theorem | sincos1sgn 11507 | The signs of the sine and cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.) |
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Theorem | sincos2sgn 11508 | The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.) |
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Theorem | sin4lt0 11509 | The sine of 4 is negative. (Contributed by Paul Chapman, 19-Jan-2008.) |
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Theorem | cos12dec 11510 | Cosine is decreasing from one to two. (Contributed by Mario Carneiro and Jim Kingdon, 6-Mar-2024.) |
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Theorem | absefi 11511 | The absolute value of the exponential of an imaginary number is one. Equation 48 of [Rudin] p. 167. (Contributed by Jason Orendorff, 9-Feb-2007.) |
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Theorem | absef 11512 | The absolute value of the exponential is the exponential of the real part. (Contributed by Paul Chapman, 13-Sep-2007.) |
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Theorem | absefib 11513 |
A complex number is real iff the exponential of its product with ![]() |
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Theorem | efieq1re 11514 | A number whose imaginary exponential is one is real. (Contributed by NM, 21-Aug-2008.) |
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Theorem | demoivre 11515 | De Moivre's Formula. Proof by induction given at http://en.wikipedia.org/wiki/De_Moivre's_formula, but restricted to nonnegative integer powers. See also demoivreALT 11516 for an alternate longer proof not using the exponential function. (Contributed by NM, 24-Jul-2007.) |
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Theorem | demoivreALT 11516 | Alternate proof of demoivre 11515. It is longer but does not use the exponential function. This is Metamath 100 proof #17. (Contributed by Steve Rodriguez, 10-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Syntax | ctau 11517 |
Extend class notation to include the constant tau, ![]() |
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Definition | df-tau 11518 |
Define the circle constant tau, ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | eirraplem 11519* | Lemma for eirrap 11520. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 5-Jan-2022.) |
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Theorem | eirrap 11520 |
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Theorem | eirr 11521 |
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Theorem | egt2lt3 11522 |
Euler's constant ![]() |
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Theorem | epos 11523 |
Euler's constant ![]() |
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Theorem | epr 11524 |
Euler's constant ![]() |
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Theorem | ene0 11525 |
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Theorem | eap0 11526 |
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Theorem | ene1 11527 |
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Theorem | eap1 11528 |
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Here we introduce elementary number theory, in particular the elementary properties of divisibility and elementary prime number theory. | ||
Syntax | cdvds 11529 | Extend the definition of a class to include the divides relation. See df-dvds 11530. |
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Definition | df-dvds 11530* | Define the divides relation, see definition in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | divides 11531* |
Define the divides relation. ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | dvdsval2 11532 | One nonzero integer divides another integer if and only if their quotient is an integer. (Contributed by Jeff Hankins, 29-Sep-2013.) |
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Theorem | dvdsval3 11533 | One nonzero integer divides another integer if and only if the remainder upon division is zero, see remark in [ApostolNT] p. 106. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 15-Jul-2014.) |
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Theorem | dvdszrcl 11534 | Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
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Theorem | nndivdvds 11535 | Strong form of dvdsval2 11532 for positive integers. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
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Theorem | nndivides 11536* | Definition of the divides relation for positive integers. (Contributed by AV, 26-Jul-2021.) |
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Theorem | dvdsdc 11537 | Divisibility is decidable. (Contributed by Jim Kingdon, 14-Nov-2021.) |
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Theorem | moddvds 11538 |
Two ways to say ![]() ![]() ![]() ![]() |
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Theorem | dvds0lem 11539 |
A lemma to assist theorems of ![]() |
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Theorem | dvds1lem 11540* |
A lemma to assist theorems of ![]() |
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Theorem | dvds2lem 11541* |
A lemma to assist theorems of ![]() |
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Theorem | iddvds 11542 | An integer divides itself. Theorem 1.1(a) in [ApostolNT] p. 14 (reflexive property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | 1dvds 11543 | 1 divides any integer. Theorem 1.1(f) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvds0 11544 | Any integer divides 0. Theorem 1.1(g) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | negdvdsb 11545 | An integer divides another iff its negation does. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdsnegb 11546 | An integer divides another iff it divides its negation. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | absdvdsb 11547 | An integer divides another iff its absolute value does. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdsabsb 11548 | An integer divides another iff it divides its absolute value. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | 0dvds 11549 | Only 0 is divisible by 0. Theorem 1.1(h) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | zdvdsdc 11550 | Divisibility of integers is decidable. (Contributed by Jim Kingdon, 17-Jan-2022.) |
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Theorem | dvdsmul1 11551 | An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdsmul2 11552 | An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | iddvdsexp 11553 | An integer divides a positive integer power of itself. (Contributed by Paul Chapman, 26-Oct-2012.) |
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Theorem | muldvds1 11554 | If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | muldvds2 11555 | If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdscmul 11556 | Multiplication by a constant maintains the divides relation. Theorem 1.1(d) in [ApostolNT] p. 14 (multiplication property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdsmulc 11557 | Multiplication by a constant maintains the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdscmulr 11558 | Cancellation law for the divides relation. Theorem 1.1(e) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdsmulcr 11559 | Cancellation law for the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | summodnegmod 11560 | The sum of two integers modulo a positive integer equals zero iff the first of the two integers equals the negative of the other integer modulo the positive integer. (Contributed by AV, 25-Jul-2021.) |
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Theorem | modmulconst 11561 | Constant multiplication in a modulo operation, see theorem 5.3 in [ApostolNT] p. 108. (Contributed by AV, 21-Jul-2021.) |
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Theorem | dvds2ln 11562 | If an integer divides each of two other integers, it divides any linear combination of them. Theorem 1.1(c) in [ApostolNT] p. 14 (linearity property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvds2add 11563 | If an integer divides each of two other integers, it divides their sum. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvds2sub 11564 | If an integer divides each of two other integers, it divides their difference. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvds2subd 11565 | Natural deduction form of dvds2sub 11564. (Contributed by Stanislas Polu, 9-Mar-2020.) |
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Theorem | dvdstr 11566 | The divides relation is transitive. Theorem 1.1(b) in [ApostolNT] p. 14 (transitive property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdsmultr1 11567 | If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.) |
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Theorem | dvdsmultr1d 11568 | Natural deduction form of dvdsmultr1 11567. (Contributed by Stanislas Polu, 9-Mar-2020.) |
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Theorem | dvdsmultr2 11569 | If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.) |
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Theorem | ordvdsmul 11570 | If an integer divides either of two others, it divides their product. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 17-Jul-2014.) |
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Theorem | dvdssub2 11571 | If an integer divides a difference, then it divides one term iff it divides the other. (Contributed by Mario Carneiro, 13-Jul-2014.) |
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Theorem | dvdsadd 11572 | An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 13-Jul-2014.) |
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Theorem | dvdsaddr 11573 | An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.) |
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Theorem | dvdssub 11574 | An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.) |
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Theorem | dvdssubr 11575 | An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.) |
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Theorem | dvdsadd2b 11576 | Adding a multiple of the base does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.) |
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Theorem | dvdslelemd 11577 | Lemma for dvdsle 11578. (Contributed by Jim Kingdon, 8-Nov-2021.) |
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Theorem | dvdsle 11578 |
The divisors of a positive integer are bounded by it. The proof does
not use ![]() |
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Theorem | dvdsleabs 11579 | The divisors of a nonzero integer are bounded by its absolute value. Theorem 1.1(i) in [ApostolNT] p. 14 (comparison property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.) |
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Theorem | dvdsleabs2 11580 | Transfer divisibility to an order constraint on absolute values. (Contributed by Stefan O'Rear, 24-Sep-2014.) |
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Theorem | dvdsabseq 11581 | If two integers divide each other, they must be equal, up to a difference in sign. Theorem 1.1(j) in [ApostolNT] p. 14. (Contributed by Mario Carneiro, 30-May-2014.) (Revised by AV, 7-Aug-2021.) |
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Theorem | dvdseq 11582 | If two nonnegative integers divide each other, they must be equal. (Contributed by Mario Carneiro, 30-May-2014.) (Proof shortened by AV, 7-Aug-2021.) |
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Theorem | divconjdvds 11583 |
If a nonzero integer ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | dvdsdivcl 11584* |
The complement of a divisor of ![]() ![]() |
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Theorem | dvdsflip 11585* | An involution of the divisors of a number. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 13-May-2016.) |
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Theorem | dvdsssfz1 11586* | The set of divisors of a number is a subset of a finite set. (Contributed by Mario Carneiro, 22-Sep-2014.) |
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Theorem | dvds1 11587 | The only nonnegative integer that divides 1 is 1. (Contributed by Mario Carneiro, 2-Jul-2015.) |
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Theorem | alzdvds 11588* | Only 0 is divisible by all integers. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdsext 11589* | Poset extensionality for division. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
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Theorem | fzm1ndvds 11590 |
No number between ![]() ![]() ![]() ![]() ![]() |
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Theorem | fzo0dvdseq 11591 |
Zero is the only one of the first ![]() ![]() |
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Theorem | fzocongeq 11592 | Two different elements of a half-open range are not congruent mod its length. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
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Theorem | addmodlteqALT 11593 | Two nonnegative integers less than the modulus are equal iff the sums of these integer with another integer are equal modulo the modulus. Shorter proof of addmodlteq 10202 based on the "divides" relation. (Contributed by AV, 14-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | dvdsfac 11594 | A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.) |
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Theorem | dvdsexp 11595 | A power divides a power with a greater exponent. (Contributed by Mario Carneiro, 23-Feb-2014.) |
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Theorem | dvdsmod 11596 |
Any number ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | mulmoddvds 11597 | 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 11598 |
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 11599 |
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 11600 | An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 11534. (Contributed by AV, 22-Jun-2021.) |
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