# Digit sum

How many are three-digit numbers that have a digit sum of 6?

Result

n =  21

#### Solution: The equations have the following integer solutions:
6 = a+b+c
a>0

##### a1=1, b1=15, c1=-10a2=1, b2=14, c2=-9a3=1, b3=13, c3=-8a4=1, b4=12, c4=-7a5=1, b5=11, c5=-6a6=1, b6=10, c6=-5a7=1, b7=9, c7=-4a8=1, b8=8, c8=-3a9=1, b9=7, c9=-2a10=1, b10=6, c10=-1a11=1, b11=5, c11=0a12=1, b12=4, c12=1a13=1, b13=3, c13=2a14=1, b14=2, c14=3a15=1, b15=1, c15=4a16=1, b16=0, c16=5a17=1, b17=-1, c17=6a18=1, b18=-2, c18=7a19=1, b19=-3, c19=8a20=1, b20=-4, c20=9a21=1, b21=-5, c21=10a22=1, b22=-6, c22=11a23=1, b23=-7, c23=12a24=1, b24=-8, c24=13a25=1, b25=-9, c25=14a26=1, b26=-10, c26=15a27=2, b27=14, c27=-10a28=2, b28=13, c28=-9a29=2, b29=12, c29=-8a30=2, b30=11, c30=-7a31=2, b31=10, c31=-6a32=2, b32=9, c32=-5a33=2, b33=8, c33=-4a34=2, b34=7, c34=-3a35=2, b35=6, c35=-2a36=2, b36=5, c36=-1a37=2, b37=4, c37=0a38=2, b38=3, c38=1a39=2, b39=2, c39=2a40=2, b40=1, c40=3a41=2, b41=0, c41=4a42=2, b42=-1, c42=5a43=2, b43=-2, c43=6a44=2, b44=-3, c44=7a45=2, b45=-4, c45=8a46=2, b46=-5, c46=9a47=2, b47=-6, c47=10a48=2, b48=-7, c48=11a49=2, b49=-8, c49=12a50=2, b50=-9, c50=13a51=2, b51=-10, c51=14a52=3, b52=13, c52=-10a53=3, b53=12, c53=-9a54=3, b54=11, c54=-8a55=3, b55=10, c55=-7a56=3, b56=9, c56=-6a57=3, b57=8, c57=-5a58=3, b58=7, c58=-4a59=3, b59=6, c59=-3a60=3, b60=5, c60=-2a61=3, b61=4, c61=-1a62=3, b62=3, c62=0a63=3, b63=2, c63=1a64=3, b64=1, c64=2a65=3, b65=0, c65=3a66=3, b66=-1, c66=4a67=3, b67=-2, c67=5a68=3, b68=-3, c68=6a69=3, b69=-4, c69=7a70=3, b70=-5, c70=8a71=3, b71=-6, c71=9a72=3, b72=-7, c72=10a73=3, b73=-8, c73=11a74=3, b74=-9, c74=12a75=3, b75=-10, c75=13a76=4, b76=12, c76=-10a77=4, b77=11, c77=-9a78=4, b78=10, c78=-8a79=4, b79=9, c79=-7a80=4, b80=8, c80=-6a81=4, b81=7, c81=-5a82=4, b82=6, c82=-4a83=4, b83=5, c83=-3a84=4, b84=4, c84=-2a85=4, b85=3, c85=-1a86=4, b86=2, c86=0a87=4, b87=1, c87=1a88=4, b88=0, c88=2a89=4, b89=-1, c89=3a90=4, b90=-2, c90=4a91=4, b91=-3, c91=5a92=4, b92=-4, c92=6a93=4, b93=-5, c93=7a94=4, b94=-6, c94=8a95=4, b95=-7, c95=9a96=4, b96=-8, c96=10a97=4, b97=-9, c97=11a98=4, b98=-10, c98=12a99=5, b99=11, c99=-10a100=5, b100=10, c100=-9a101=5, b101=9, c101=-8a102=5, b102=8, c102=-7a103=5, b103=7, c103=-6a104=5, b104=6, c104=-5a105=5, b105=5, c105=-4a106=5, b106=4, c106=-3a107=5, b107=3, c107=-2a108=5, b108=2, c108=-1a109=5, b109=1, c109=0a110=5, b110=0, c110=1a111=5, b111=-1, c111=2a112=5, b112=-2, c112=3a113=5, b113=-3, c113=4a114=5, b114=-4, c114=5a115=5, b115=-5, c115=6a116=5, b116=-6, c116=7a117=5, b117=-7, c117=8a118=5, b118=-8, c118=9a119=5, b119=-9, c119=10a120=5, b120=-10, c120=11a121=6, b121=10, c121=-10a122=6, b122=9, c122=-9a123=6, b123=8, c123=-8a124=6, b124=7, c124=-7a125=6, b125=6, c125=-6a126=6, b126=5, c126=-5a127=6, b127=4, c127=-4a128=6, b128=3, c128=-3a129=6, b129=2, c129=-2a130=6, b130=1, c130=-1a131=6, b131=0, c131=0a132=6, b132=-1, c132=1a133=6, b133=-2, c133=2a134=6, b134=-3, c134=3a135=6, b135=-4, c135=4a136=6, b136=-5, c136=5a137=6, b137=-6, c137=6a138=6, b138=-7, c138=7a139=6, b139=-8, c139=8a140=6, b140=-9, c140=9a141=6, b141=-10, c141=10a142=7, b142=9, c142=-10a143=7, b143=8, c143=-9a144=7, b144=7, c144=-8a145=7, b145=6, c145=-7a146=7, b146=5, c146=-6a147=7, b147=4, c147=-5a148=7, b148=3, c148=-4a149=7, b149=2, c149=-3a150=7, b150=1, c150=-2a151=7, b151=0, c151=-1a152=7, b152=-1, c152=0a153=7, b153=-2, c153=1a154=7, b154=-3, c154=2a155=7, b155=-4, c155=3a156=7, b156=-5, c156=4a157=7, b157=-6, c157=5a158=7, b158=-7, c158=6a159=7, b159=-8, c159=7a160=7, b160=-9, c160=8a161=7, b161=-10, c161=9a162=8, b162=8, c162=-10a163=8, b163=7, c163=-9a164=8, b164=6, c164=-8a165=8, b165=5, c165=-7a166=8, b166=4, c166=-6a167=8, b167=3, c167=-5a168=8, b168=2, c168=-4a169=8, b169=1, c169=-3a170=8, b170=0, c170=-2a171=8, b171=-1, c171=-1a172=8, b172=-2, c172=0a173=8, b173=-3, c173=1a174=8, b174=-4, c174=2a175=8, b175=-5, c175=3a176=8, b176=-6, c176=4a177=8, b177=-7, c177=5a178=8, b178=-8, c178=6a179=8, b179=-9, c179=7a180=8, b180=-10, c180=8a181=9, b181=7, c181=-10a182=9, b182=6, c182=-9a183=9, b183=5, c183=-8a184=9, b184=4, c184=-7a185=9, b185=3, c185=-6a186=9, b186=2, c186=-5a187=9, b187=1, c187=-4a188=9, b188=0, c188=-3a189=9, b189=-1, c189=-2a190=9, b190=-2, c190=-1a191=9, b191=-3, c191=0a192=9, b192=-4, c192=1a193=9, b193=-5, c193=2a194=9, b194=-6, c194=3a195=9, b195=-7, c195=4a196=9, b196=-8, c196=5a197=9, b197=-9, c197=6a198=9, b198=-10, c198=7a199=10, b199=6, c199=-10a200=10, b200=5, c200=-9a201=10, b201=4, c201=-8a202=10, b202=3, c202=-7a203=10, b203=2, c203=-6a204=10, b204=1, c204=-5a205=10, b205=0, c205=-4a206=10, b206=-1, c206=-3a207=10, b207=-2, c207=-2a208=10, b208=-3, c208=-1a209=10, b209=-4, c209=0a210=10, b210=-5, c210=1a211=10, b211=-6, c211=2a212=10, b212=-7, c212=3a213=10, b213=-8, c213=4a214=10, b214=-9, c214=5a215=10, b215=-10, c215=6a216=11, b216=5, c216=-10a217=11, b217=4, c217=-9a218=11, b218=3, c218=-8a219=11, b219=2, c219=-7a220=11, b220=1, c220=-6a221=11, b221=0, c221=-5a222=11, b222=-1, c222=-4a223=11, b223=-2, c223=-3a224=11, b224=-3, c224=-2a225=11, b225=-4, c225=-1a226=11, b226=-5, c226=0a227=11, b227=-6, c227=1a228=11, b228=-7, c228=2a229=11, b229=-8, c229=3a230=11, b230=-9, c230=4a231=11, b231=-10, c231=5a232=12, b232=4, c232=-10a233=12, b233=3, c233=-9a234=12, b234=2, c234=-8a235=12, b235=1, c235=-7a236=12, b236=0, c236=-6a237=12, b237=-1, c237=-5a238=12, b238=-2, c238=-4a239=12, b239=-3, c239=-3a240=12, b240=-4, c240=-2a241=12, b241=-5, c241=-1a242=12, b242=-6, c242=0a243=12, b243=-7, c243=1a244=12, b244=-8, c244=2a245=12, b245=-9, c245=3a246=12, b246=-10, c246=4a247=13, b247=3, c247=-10a248=13, b248=2, c248=-9a249=13, b249=1, c249=-8a250=13, b250=0, c250=-7a251=13, b251=-1, c251=-6a252=13, b252=-2, c252=-5a253=13, b253=-3, c253=-4a254=13, b254=-4, c254=-3a255=13, b255=-5, c255=-2a256=13, b256=-6, c256=-1a257=13, b257=-7, c257=0a258=13, b258=-8, c258=1a259=13, b259=-9, c259=2a260=13, b260=-10, c260=3a261=14, b261=2, c261=-10a262=14, b262=1, c262=-9a263=14, b263=0, c263=-8a264=14, b264=-1, c264=-7a265=14, b265=-2, c265=-6a266=14, b266=-3, c266=-5a267=14, b267=-4, c267=-4a268=14, b268=-5, c268=-3a269=14, b269=-6, c269=-2a270=14, b270=-7, c270=-1a271=14, b271=-8, c271=0a272=14, b272=-9, c272=1a273=14, b273=-10, c273=2a274=15, b274=1, c274=-10a275=15, b275=0, c275=-9a276=15, b276=-1, c276=-8a277=15, b277=-2, c277=-7a278=15, b278=-3, c278=-6a279=15, b279=-4, c279=-5a280=15, b280=-5, c280=-4a281=15, b281=-6, c281=-3a282=15, b282=-7, c282=-2a283=15, b283=-8, c283=-1a284=15, b284=-9, c284=0a285=15, b285=-10, c285=1a286=16, b286=0, c286=-10a287=16, b287=-1, c287=-9a288=16, b288=-2, c288=-8a289=16, b289=-3, c289=-7a290=16, b290=-4, c290=-6a291=16, b291=-5, c291=-5a292=16, b292=-6, c292=-4a293=16, b293=-7, c293=-3a294=16, b294=-8, c294=-2a295=16, b295=-9, c295=-1a296=16, b296=-10, c296=0a297=17, b297=-1, c297=-10a298=17, b298=-2, c298=-9a299=17, b299=-3, c299=-8a300=17, b300=-4, c300=-7a301=17, b301=-5, c301=-6a302=17, b302=-6, c302=-5a303=17, b303=-7, c303=-4a304=17, b304=-8, c304=-3a305=17, b305=-9, c305=-2a306=17, b306=-10, c306=-1a307=18, b307=-2, c307=-10a308=18, b308=-3, c308=-9a309=18, b309=-4, c309=-8a310=18, b310=-5, c310=-7a311=18, b311=-6, c311=-6a312=18, b312=-7, c312=-5a313=18, b313=-8, c313=-4a314=18, b314=-9, c314=-3a315=18, b315=-10, c315=-2a316=19, b316=-3, c316=-10a317=19, b317=-4, c317=-9a318=19, b318=-5, c318=-8a319=19, b319=-6, c319=-7a320=19, b320=-7, c320=-6a321=19, b321=-8, c321=-5a322=19, b322=-9, c322=-4a323=19, b323=-10, c323=-3a324=20, b324=-4, c324=-10a325=20, b325=-5, c325=-9a326=20, b326=-6, c326=-8a327=20, b327=-7, c327=-7a328=20, b328=-8, c328=-6a329=20, b329=-9, c329=-5a330=20, b330=-10, c330=-4a331=21, b331=-5, c331=-10a332=21, b332=-6, c332=-9a333=21, b333=-7, c333=-8a334=21, b334=-8, c334=-7a335=21, b335=-9, c335=-6a336=21, b336=-10, c336=-5a337=22, b337=-6, c337=-10a338=22, b338=-7, c338=-9a339=22, b339=-8, c339=-8a340=22, b340=-9, c340=-7a341=22, b341=-10, c341=-6a342=23, b342=-7, c342=-10a343=23, b343=-8, c343=-9a344=23, b344=-9, c344=-8a345=23, b345=-10, c345=-7a346=24, b346=-8, c346=-10a347=24, b347=-9, c347=-9a348=24, b348=-10, c348=-8a349=25, b349=-9, c349=-10a350=25, b350=-10, c350=-9a351=26, b351=-10, c351=-10

Number of solutions found: 351

Calculated by our Diofant problems and integer equations.

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