1414-1515-1616-From-5757-units-How-many-of-them-are-divided-by-13-and-the-remainder-is-10-

Question Number 211147 by hardmath last updated on 29/Aug/24

$$ \\ $$1414; 1515; 1616;…; From 5757 units

How many of them are divided by 13 and the remainder is 10?

Answered by mm1342 last updated on 30/Aug/24

−a−4b−3c+d≡^(13) 10 ⇒a+4b+3c−d≡^(13) 3 for “ abab ” if abab≡^(13) 10⇒ 4a+3b≡^(13) 3 ; 1≤a≤5 ⇒4a+3b=3 or 16 or 29 or 42 ... a=1⇒3b=12⇒b=4⇒1414 ✓ a=2⇒3b=21⇒b=7⇒2727 ✓ a=4⇒3b=0⇒b=0⇒4040 ✓ a=5⇒3b=9⇒b=3⇒5353 ✓

$$−{a}−\mathrm{4}{b}−\mathrm{3}{c}+{d}\overset{\mathrm{13}} {\equiv}\mathrm{10} \\ $$$$\Rightarrow{a}+\mathrm{4}{b}+\mathrm{3}{c}−{d}\overset{\mathrm{13}} {\equiv}\mathrm{3} \\ $$$${for}\:\:“\:{abab}\:''\:{if}\:\:{abab}\overset{\mathrm{13}} {\equiv}\mathrm{10}\Rightarrow\:\mathrm{4}{a}+\mathrm{3}{b}\overset{\mathrm{13}} {\equiv}\mathrm{3}\:\:;\:\mathrm{1}\leqslant{a}\leqslant\mathrm{5} \\ $$$$\Rightarrow\mathrm{4}{a}+\mathrm{3}{b}=\mathrm{3}\:{or}\:\mathrm{16}\:{or}\:\mathrm{29}\:{or}\:\mathrm{42}\:\:… \\ $$$${a}=\mathrm{1}\Rightarrow\mathrm{3}{b}=\mathrm{12}\Rightarrow{b}=\mathrm{4}\Rightarrow\mathrm{1414}\:\checkmark \\ $$$${a}=\mathrm{2}\Rightarrow\mathrm{3}{b}=\mathrm{21}\Rightarrow{b}=\mathrm{7}\Rightarrow\mathrm{2727}\:\:\checkmark \\ $$$${a}=\mathrm{4}\Rightarrow\mathrm{3}{b}=\mathrm{0}\Rightarrow{b}=\mathrm{0}\Rightarrow\mathrm{4040}\:\:\:\checkmark \\ $$$${a}=\mathrm{5}\Rightarrow\mathrm{3}{b}=\mathrm{9}\Rightarrow{b}=\mathrm{3}\Rightarrow\mathrm{5353}\:\:\:\checkmark \\ $$$$ \\ $$

Commented by hardmath last updated on 30/Aug/24