find the least no which when divided by 17, 16 ,15 leaves the remainder 3, 10, 0 respectively u have to explain the step

1 Answer | Add Yours

Top Answer

embizze's profile pic

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

Find the least number such that the remainder when divided by 17 is 3, when divided by 16 is 10, and when divided by 15 is 0.

If I have the question correctly stated, then the answer from your book is wrong since 3910/15 is not a whole number.

(1) The number must be a multiple of 15 since the remainder is 0.

(2) We look for the first multiple of 15 that is 10 more than a multiple of 16. We find that 90=15*6 and 16*5+10=90. The LCM of 15 and 16 is 240, so the answer will be a number of the form 90+240x.

(3) We need a number of the form 90+240x that is 3 more than a multiple of 17. We notice that 90=17*5+5, 330=17*19+7 so we try the other direction and see that -150=17*(-9)+3. We presumably want the smallest positive number or the problem is unanswerable, so we add the LCM of 240 and 17 which is 4080 to get 3930.

Note that 3930=15*262, 3930=16*245+10, and 3930=17*231+3 so 3930 is the solution.

 

If you are studying modular arithmetic, note that the number x has the following properties:

x=0(mod15)
x=3(mod17)
x=10(mod16)

So we are looking for a w such that

15w=10(mod16) or w=6(mod16)
15w=3(mod17)   or 5w=1(mod17)
Solving for w yields w=262 and 15(262)=3930, the number we are seeking.

 

 

We’ve answered 318,915 questions. We can answer yours, too.

Ask a question