Question

Găseste numerele naturale de patru cifre care au suma cifrelor 9

Answer 1

Salutare !!!

Pentru a rezolva cât mai ușor problema vom nota cu abcd numerele naturale de patru cifre căutate

Ne înarmăm cu răbdare, bem o ceașcă de calmitate și începem să scriem

a,b,c,d - cifre; a,b,c,d∈{0,1,2,3,4,5,6,7,8,9}; a ≠ 0

a+b+c+d = 9

Enunțul problemei NU precizează că numerele au cifrele distincte sau cifrele sunt diferite doua câte două. Analizam ce valoare poate lua a (9 cazuri) apoi ce valoare poate lua b

• a = 1  ⇒ b+c+d =8

b = 0 ⇒ c+d=8 ⇒ cd∈{08,80,17,71,62,26,35,53,44}

abcd ∈ {1008,1080,1017,1071,1062,1026,1035,1053,1044} -9 nr

b = 1⇒c+d=7⇒cd∈{07,70,16,61,25,52,34,43}

abcd∈{1107, 1170, 1116, 1161, 1125, 1152, 1134, 1143}  -8 nr

b = 2⇒c+d=6⇒cd∈{06,60,15,51,24,42,33}

abcd∈{1206, 1260, 1251, 1215, 1224, 1242, 1233}  -7 nr

b = 3⇒c+d=5⇒cd∈{05,50,41,14,23,32}

abcd∈{1305,1350,1314,1341,1323,1332}  -6 nr

b = 4⇒c+d=4⇒cd∈{04,40,13,31,22}⇒abcd∈{1404,1440,1413,1431,1422} -5 nr

b = 5⇒c+d=3⇒ cd∈{03,30,12,21}⇒abcd∈{1503,1530,1512,1521} -4 nr

b = 6 ⇒c+d=2⇒ cd∈{02,20,11}⇒abcd∈{1602,1620,1611} -3 nr

b = 7 ⇒c+d=1⇒ cd∈{01,10}⇒abcd∈{1701,1710} -2 nr

b = 8⇒c+d=0⇒cd=00 ⇒ abcd = 1800 -1 nr

• a = 2  ⇒ b+c+d =7

b = 0 ⇒c+d=7⇒cd∈{07,70,16,61,25,52,34,43}

abcd∈{2007,2070,2016,2061,2025,2052,2034,2043}  -8 nr

b = 1⇒c+d=6⇒cd∈{06,60,15,51,24,42,33}

abcd∈{2106, 2160, 2151, 2115, 2124, 2142, 2133}  -7 nr

b = 2⇒c+d=5⇒cd∈{05,50,41,14,23,32}

abcd∈{2205, 2250, 2214, 2241, 2223, 2232} -6 nr

b = 3⇒c+d=4⇒cd∈{04,40,13,31,22}⇒abcd∈{2304,2340,2313,2331,2322}

b = 4⇒c+d=3⇒cd∈{03,30,12,21}⇒abcd∈{2403,2430,2412,2421}  -4 nr

b = 5⇒c+d=2⇒cd∈{02,20,11}⇒abcd∈{2502,2520,2511} -3 nr

b = 6⇒c+d=1⇒cd∈{01,10}⇒abcd∈{2601,2610} -2 nr

b = 7⇒c+d=0⇒cd=00 ⇒ abcd = 2700 -1 nr

• a = 3  ⇒ b+c+d =6

b = 0 ⇒c+d=6⇒cd∈{06,60,15,51,24,42,33}

abcd∈{3006,3060,3015,3051,3024,3042,3033}  -7 nr

b=1⇒c+d=5⇒cd∈{05,50,41,14,23,32}

abcd∈{3105,3150,3141,3114,3123,3132}  -6 nr

b=2⇒c+d=4⇒cd∈{04,40,13,31,22}⇒abcd∈{3204,3240,3213,3231,3222}

b=3⇒c+d=3⇒cd∈{03,30,12,21}⇒abcd∈{3303,3330,3312,3321} -4 nr

b = 4⇒c+d=2⇒cd∈{02,20,11}⇒abcd∈{3402,3420,3411} -3 nr

b = 5⇒c+d=1⇒cd∈{01,10}⇒abcd∈{3501,3510} -2 nr

b = 6⇒c+d=0⇒cd=00 ⇒ abcd = 3600 -1 nr

• a = 4  ⇒ b+c+d =5

b = 0⇒c+d=5⇒cd∈{05,50,41,14,23,32}

abcd∈{4005,4050,4014,4041,4023,4032}  -6 nr

b = 1⇒c+d=4⇒cd∈{04,40,13,31,22}⇒abcd∈{4104,4140,4113,4131,4122}

b = 2⇒c+d=3⇒cd∈{03,30,12,21}⇒abcd∈{4203,4230,4212,4221} -4 nr

b = 3⇒c+d=2⇒cd∈{02,20,11}⇒abcd∈{4302,4320,4311} -3 nr

b = 4⇒c+d=1⇒cd∈{01,10}⇒abcd∈{4401,4410} -2 nr

b = 5⇒c+d=0⇒cd=00 ⇒ abcd = 4500 -1 nr

• a = 5  ⇒ b+c+d =4

b=0⇒c+d=4⇒cd∈{04,40,13,31,22}⇒abcd∈{5040,5004,5013,5031,5022}

b = 1⇒c+d=3⇒cd∈{03,30,12,21}⇒abcd∈{5103,5130,5112,5121} -4 nr

b = 2⇒c+d=2⇒cd∈{02,20,11}⇒abcd∈{5202,5220,5211} -3 nr

b = 3⇒c+d=1⇒cd∈{01,10}⇒abcd∈{5301,5310} -2 nr

b = 4⇒c+d=0⇒cd=00 ⇒ abcd = 5400 -1 nr

• a = 6  ⇒ b+c+d =3

b = 0⇒c+d=3⇒cd∈{03,30,12,21}⇒abcd∈{6003,6030,6012,6021} -4 nr

b = 1⇒c+d=2⇒cd∈{02,20,11}⇒abcd∈{6102,6120,6111} -3 nr

b = 2⇒c+d=1⇒cd∈{01,10}⇒abcd∈{6201,6210} -2 nr

b = 3⇒c+d=0⇒cd=00 ⇒ abcd = 6300 -1 nr

• a = 7  ⇒ b+c+d=2

b = 0⇒c+d=2⇒cd∈{02,20,11}⇒abcd∈{7002,7020,7011} -3 nr

b = 1⇒c+d=1⇒cd∈{01,10}⇒abcd∈{7101,7110} -2 nr

b = 2⇒c+d=0⇒cd=00 ⇒ abcd = 7200 -1 nr

• a = 8  ⇒ b+c+d=1

b = 0⇒c+d=1⇒cd∈{01,10}⇒abcd∈{8001,8010} -2 nr

b = 1⇒c+d=0⇒cd=00 ⇒ abcd = 8100 -1 nr

• a = 9  ⇒ b+c+d = 0

b = 0⇒c+d=0⇒cd=00 ⇒ abcd = 9000 -1 nr

Vei observa că în funcție de ce valori ia a avem un anumit număr de numere ce respectă cerințele problemei:

a = 1 avem 45 numere;

a = 2 avem 36 numere;

a=3 avem 28 numere;

a = 4 avem 21 numere;

a = 5 avem 15 numere;

a = 6 avem 10 numere;

a = 7 avem 6 numere;

a = 8 avem 3 numere;

a = 9 avem 1 număr

Total numere: 45+36+28+21+15+10+6+3+1 = 165 de numere de patru cifre care au suma cifrelor 9