Question

Find the greatest number that will divide 446, 574 and 704 to leave the remainders 5, 7 and 11 respectively

Answer 1

$\huge\mathbb{SOLUTION:-}$

• Subtract the remainders leave by the numbers

$\sf\bullet 446 -5= 441\\ \\ \sf\bullet 574-7= 567\\ \\ \sf\bullet 704-11= 693$

◆Now find the H.C.F by Euclid division lemma

$\sf\bullet H.C.F\:of\: 441\:and\:567\\ \\ \sf\implies 567= 441\times 1+126\\ \\ \sf\implies 441=126\times 3 +63\\ \\ \sf\implies 126= 63\times 2+0\\ \\ \sf\bullet {\purple{H.C.F \:if\:441\:and\:567 =63}}$

• Now H.C.F of 441 and 693

$\sf\implies 693=441\times 1+252\\ \\ \sf\implies 441=252\times 1+189\\ \\ \sf\implies 252=189\times 1+63\\ \\ \sf\implies 189= 63\times 3+0\\ \\ \sf\bullet {\blue{H.C.F\:of\:441\:and\:693= 63}}$

• So the number which divides

446 ,574 and 704 and leaves remainder 5,7, and 11 is

$\boxed{\sf{63}}$

Answer 2

$\huge\mathcal{ \overbrace{ \underbrace{ \pink{ \fbox{ \green{ \blue{A} \pink{n} \red{s} \green{w} \purple{e} \blue{r}}}}}}}$

$\bf{We \: have \:to \: subtract \: the \: remainders \: leave \: by} \\ \bf{ \ the \: numbers. } \\ \bf{∴446 - 5 = 441} \\ ∴ \bf{574 - 7 = 567} \\ ∴ \bf{704 - 11 = 693} \\ \\ \bf{Now ,\: finding \: the \:H.C.F. \:by \: Euclid's \: division} \\ \bf{lemma.} \\ ∴ \bf{H.C.F. \: of \: 441 \: and \: 567} \\ \bf{⇒567 = 441 \times 1 + 126 } \\ ⇒ \bf{441 = 126 \times 3 + 63} \\ ⇒ \bf{126 = 63 \times 2 + 0 } \\ \tt{ \red{∴H.C.F. \: of \: 441 \: and \: 567 = 63}} \\ \\ ∴ \bf{H.C.F. \: of \: 441 \: and \: 693 } \\ ⇒ \bf{693 = 441 \times 1 + 252} \\ ⇒ \bf{441 = 252 \times 1 + 189} \\ ⇒ \bf{252 = 189 \times 1 + 63} \\ ⇒ \bf{189 = 63 \times 3 + 0} \\ \tt{ \red{∴ H.C.F. \: of \:441 \: and \: 693 = 63 }} \\ \\ \bf{So, \: the \: number \: which \: divides \:446, \: 574 \: and \: 704 } \\ \bf{leaving \: a \: remainder \: 5,\:7 \: and \: 11 \: is} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:\huge{ \boxed {\boxed{ \purple{ \mathfrak{63}}}}}$