Divisibility

Number Theory Level 3

F i n d t h e s m a l l e s t n o w h i c h w h e n d i v i d e d b y 35 l e a v e s a r e m a i n d e r 25 , w i t h 45 l e a v e s 35 a s a r e m a n d w i t h 55 l e a v e s 45. Find\quad the\quad smallest\quad no\quad which\quad when\quad \\ divided\quad by\quad 35\quad leaves\quad a\quad remainder\\ 25,\quad with\quad 45\quad leaves\quad 35\quad as\quad a\quad rem\\ \quad \quad and\quad with\quad 55\quad leaves\quad 45.


The answer is 3455.

Rajdeep Dhingra by Rajdeep Dhingra , 20, India

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2 solutions

Sunil Pradhan
Oct 22, 2014

No ÷ 35 remainder 25

No ÷ 45 remainder 35

No ÷ 55 remainder 45

here the difference in divisor and remainder (35 – 25), (45 – 35) and (55 – 45) = 10 constant

so least number = LCM of divisors – difference

required least number = (LCM of 35, 45, 55) – 10

= 3455

3 Helpful 0 Interesting 0 Brilliant 0 Confused
Rajdeep Dhingra
Oct 16, 2014

L e t t h a t n u m b e r b e x . T h e n x = 25 m o d 35 x = 35 m o d 45 x = 45 m o d 55 S o , i t c a n b e r e w r i t t e n a s x = − 10 m o d 35 x = − 10 m o d 45 x = − 10 m o d 55 H e n c e t h e a n s w e r i s 3455 i e . 10 l e s s t h a n 3465 ( L C M o f 35 , 45 , 55 ) Let\quad that\quad number\quad be\quad x.\\ Then\\ x\quad =\quad 25\quad mod\quad 35\\ x\quad =\quad 35\quad mod\quad 45\\ x\quad =\quad 45\quad mod\quad 55\\ So\quad ,\quad it\quad can\quad be\quad rewritten\quad as\\ x\quad =\quad -10\quad mod\quad 35\\ x\quad =\quad -10\quad mod\quad 45\\ x\quad =\quad -10\quad mod\quad 55\\ Hence\quad the\quad answer\quad is\quad 3455\quad ie.\quad 10\quad less\\ than\quad 3465\quad (LCM\quad of\quad 35,45,55)

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Can you explain how you got -10

Archit Boobna - 6 years, 8 months ago

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25 mod 35 can be re written as 25-35 mod 35 which is -10 mod 35 this is a property of mod function. also when we divide something with 35 and we get 25 as remainder we can increase the quotient by 1 and subtract 35 from 25.

Rajdeep Dhingra - 6 years, 8 months ago

rajdeep bhaiya mod function padhne ke liye kaun si book se padhe koi specific book hai ya fir kisi website se padhe .

Real Champ - 3 years, 11 months ago

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