# Digit sum

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

Correct result:

n =  21

#### Solution:

$n=21$

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

Number of solutions found: 210
##### a1=1, b1=-7, c1=12a2=1, b2=-6, c2=11a3=1, b3=-5, c3=10a4=1, b4=-4, c4=9a5=1, b5=-3, c5=8a6=1, b6=-2, c6=7a7=1, b7=-1, c7=6a8=1, b8=0, c8=5a9=1, b9=1, c9=4a10=1, b10=2, c10=3a11=1, b11=3, c11=2a12=1, b12=4, c12=1a13=1, b13=5, c13=0a14=1, b14=6, c14=-1a15=1, b15=7, c15=-2a16=1, b16=8, c16=-3a17=1, b17=9, c17=-4a18=1, b18=10, c18=-5a19=1, b19=11, c19=-6a20=1, b20=12, c20=-7a21=2, b21=-7, c21=11a22=2, b22=-6, c22=10a23=2, b23=-5, c23=9a24=2, b24=-4, c24=8a25=2, b25=-3, c25=7a26=2, b26=-2, c26=6a27=2, b27=-1, c27=5a28=2, b28=0, c28=4a29=2, b29=1, c29=3a30=2, b30=2, c30=2a31=2, b31=3, c31=1a32=2, b32=4, c32=0a33=2, b33=5, c33=-1a34=2, b34=6, c34=-2a35=2, b35=7, c35=-3a36=2, b36=8, c36=-4a37=2, b37=9, c37=-5a38=2, b38=10, c38=-6a39=2, b39=11, c39=-7a40=3, b40=-7, c40=10a41=3, b41=-6, c41=9a42=3, b42=-5, c42=8a43=3, b43=-4, c43=7a44=3, b44=-3, c44=6a45=3, b45=-2, c45=5a46=3, b46=-1, c46=4a47=3, b47=0, c47=3a48=3, b48=1, c48=2a49=3, b49=2, c49=1a50=3, b50=3, c50=0a51=3, b51=4, c51=-1a52=3, b52=5, c52=-2a53=3, b53=6, c53=-3a54=3, b54=7, c54=-4a55=3, b55=8, c55=-5a56=3, b56=9, c56=-6a57=3, b57=10, c57=-7a58=4, b58=-7, c58=9a59=4, b59=-6, c59=8a60=4, b60=-5, c60=7a61=4, b61=-4, c61=6a62=4, b62=-3, c62=5a63=4, b63=-2, c63=4a64=4, b64=-1, c64=3a65=4, b65=0, c65=2a66=4, b66=1, c66=1a67=4, b67=2, c67=0a68=4, b68=3, c68=-1a69=4, b69=4, c69=-2a70=4, b70=5, c70=-3a71=4, b71=6, c71=-4a72=4, b72=7, c72=-5a73=4, b73=8, c73=-6a74=4, b74=9, c74=-7a75=5, b75=-7, c75=8a76=5, b76=-6, c76=7a77=5, b77=-5, c77=6a78=5, b78=-4, c78=5a79=5, b79=-3, c79=4a80=5, b80=-2, c80=3a81=5, b81=-1, c81=2a82=5, b82=0, c82=1a83=5, b83=1, c83=0a84=5, b84=2, c84=-1a85=5, b85=3, c85=-2a86=5, b86=4, c86=-3a87=5, b87=5, c87=-4a88=5, b88=6, c88=-5a89=5, b89=7, c89=-6a90=5, b90=8, c90=-7a91=6, b91=-7, c91=7a92=6, b92=-6, c92=6a93=6, b93=-5, c93=5a94=6, b94=-4, c94=4a95=6, b95=-3, c95=3a96=6, b96=-2, c96=2a97=6, b97=-1, c97=1a98=6, b98=0, c98=0a99=6, b99=1, c99=-1a100=6, b100=2, c100=-2a101=6, b101=3, c101=-3a102=6, b102=4, c102=-4a103=6, b103=5, c103=-5a104=6, b104=6, c104=-6a105=6, b105=7, c105=-7a106=7, b106=-7, c106=6a107=7, b107=-6, c107=5a108=7, b108=-5, c108=4a109=7, b109=-4, c109=3a110=7, b110=-3, c110=2a111=7, b111=-2, c111=1a112=7, b112=-1, c112=0a113=7, b113=0, c113=-1a114=7, b114=1, c114=-2a115=7, b115=2, c115=-3a116=7, b116=3, c116=-4a117=7, b117=4, c117=-5a118=7, b118=5, c118=-6a119=7, b119=6, c119=-7a120=8, b120=-7, c120=5a121=8, b121=-6, c121=4a122=8, b122=-5, c122=3a123=8, b123=-4, c123=2a124=8, b124=-3, c124=1a125=8, b125=-2, c125=0a126=8, b126=-1, c126=-1a127=8, b127=0, c127=-2a128=8, b128=1, c128=-3a129=8, b129=2, c129=-4a130=8, b130=3, c130=-5a131=8, b131=4, c131=-6a132=8, b132=5, c132=-7a133=9, b133=-7, c133=4a134=9, b134=-6, c134=3a135=9, b135=-5, c135=2a136=9, b136=-4, c136=1a137=9, b137=-3, c137=0a138=9, b138=-2, c138=-1a139=9, b139=-1, c139=-2a140=9, b140=0, c140=-3a141=9, b141=1, c141=-4a142=9, b142=2, c142=-5a143=9, b143=3, c143=-6a144=9, b144=4, c144=-7a145=10, b145=-7, c145=3a146=10, b146=-6, c146=2a147=10, b147=-5, c147=1a148=10, b148=-4, c148=0a149=10, b149=-3, c149=-1a150=10, b150=-2, c150=-2a151=10, b151=-1, c151=-3a152=10, b152=0, c152=-4a153=10, b153=1, c153=-5a154=10, b154=2, c154=-6a155=10, b155=3, c155=-7a156=11, b156=-7, c156=2a157=11, b157=-6, c157=1a158=11, b158=-5, c158=0a159=11, b159=-4, c159=-1a160=11, b160=-3, c160=-2a161=11, b161=-2, c161=-3a162=11, b162=-1, c162=-4a163=11, b163=0, c163=-5a164=11, b164=1, c164=-6a165=11, b165=2, c165=-7a166=12, b166=-7, c166=1a167=12, b167=-6, c167=0a168=12, b168=-5, c168=-1a169=12, b169=-4, c169=-2a170=12, b170=-3, c170=-3a171=12, b171=-2, c171=-4a172=12, b172=-1, c172=-5a173=12, b173=0, c173=-6a174=12, b174=1, c174=-7a175=13, b175=-7, c175=0a176=13, b176=-6, c176=-1a177=13, b177=-5, c177=-2a178=13, b178=-4, c178=-3a179=13, b179=-3, c179=-4a180=13, b180=-2, c180=-5a181=13, b181=-1, c181=-6a182=13, b182=0, c182=-7a183=14, b183=-7, c183=-1a184=14, b184=-6, c184=-2a185=14, b185=-5, c185=-3a186=14, b186=-4, c186=-4a187=14, b187=-3, c187=-5a188=14, b188=-2, c188=-6a189=14, b189=-1, c189=-7a190=15, b190=-7, c190=-2a191=15, b191=-6, c191=-3a192=15, b192=-5, c192=-4a193=15, b193=-4, c193=-5a194=15, b194=-3, c194=-6a195=15, b195=-2, c195=-7a196=16, b196=-7, c196=-3a197=16, b197=-6, c197=-4a198=16, b198=-5, c198=-5a199=16, b199=-4, c199=-6a200=16, b200=-3, c200=-7a201=17, b201=-7, c201=-4a202=17, b202=-6, c202=-5a203=17, b203=-5, c203=-6a204=17, b204=-4, c204=-7a205=18, b205=-7, c205=-5a206=18, b206=-6, c206=-6a207=18, b207=-5, c207=-7a208=19, b208=-7, c208=-6a209=19, b209=-6, c209=-7a210=20, b210=-7, c210=-7

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