Difference between revisions of "Divisibility rules/Rule 1 for 7 proof"
Thurinalas (talk | contribs) m (→Proof) |
m (Fixed) |
||
(One intermediate revision by the same user not shown) | |||
Line 2: | Line 2: | ||
''This is based on [[Divisibility_rules/Rule_for_11_proof | divisibility by 11 rule ]]'' | ''This is based on [[Divisibility_rules/Rule_for_11_proof | divisibility by 11 rule ]]'' | ||
− | Assume N has 3k digits, otherwise add zeros to the left. | + | Assume <math>N</math> has <math>3k</math> digits, otherwise add zeros to the left. |
[[Without_loss_of_generality | WLOG ]] let <math>N = a_{3k-1}a_{3k-2}a_{3k-3}\cdots a_8a_7a_6a_5a_4a_3a_2a_1a_0</math> where the <math>a_i</math> are [[base numbers | base-ten]] numbers. | [[Without_loss_of_generality | WLOG ]] let <math>N = a_{3k-1}a_{3k-2}a_{3k-3}\cdots a_8a_7a_6a_5a_4a_3a_2a_1a_0</math> where the <math>a_i</math> are [[base numbers | base-ten]] numbers. | ||
− | Then < | + | Then <cmath>N = 10^{3k-1} a_{3k-1} + 10^{3k-2} a_{3k-2} + 10^{3k-3} a_{3k-3} \cdots + 10^8 a_8 + 10^7 a_7 + 10^6 a_6 + 10^5 a_5 + 10^4 a_4 + 10^3 a_3 + 10^2 a_2 + 10 a_1 + a_0</cmath> |
− | < | + | <cmath>= 1000^{k-1} (100 a_{3k-1} + 10 a_{3k-2} + a_{3k-3}) \cdots + 1000^2 (100 a_8 + 10 a_7 + a_6) + 1000 (100 a_5 + 10 a_4 + a_3) + (100 a_2 + 10 a_1 + a_0).</cmath> |
Rewriting or partitioning <math>N</math> into 3 digit numbers (<math>a_{3k-1}a_{3k-2}a_{3k-3}\cdots a_8a_7a_6 a_5a_4a_3 a_2a_1a_0</math>). | Rewriting or partitioning <math>N</math> into 3 digit numbers (<math>a_{3k-1}a_{3k-2}a_{3k-3}\cdots a_8a_7a_6 a_5a_4a_3 a_2a_1a_0</math>). |
Latest revision as of 15:07, 2 October 2024
Proof
This is based on divisibility by 11 rule
Assume has digits, otherwise add zeros to the left. WLOG let where the are base-ten numbers. Then
Rewriting or partitioning into 3 digit numbers ().
This is the alternating sum of groups of 3 digit numbers of , which is what we wanted.