Odd patterns

Number Theory Level 1

What is the final digit of 1 1 × 3 3 × 5 5 × 7 7 × ⋯ × 201 5 2015 1^1\times3^3\times5^5\times7^7\times\cdots \times2015^{2015} ?

3 1 5 7 9

Rob Matuschek by Rob Matuschek , 54, USA

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

9 solutions

Otto Bretscher
Sep 10, 2015

An odd multiple of 5 ends in a 5 \boxed{5}

39 Helpful 0 Interesting 0 Brilliant 0 Confused

Eloquent indeed.

Rob Matuschek - 5 years, 9 months ago

Thought of the ans in the same way.

Arunit Mazumdar - 5 years, 9 months ago

can u solve the question kindly? :)

Princess iqra - 5 years, 8 months ago

Log in to reply

Factoring out a 5, we see that the given number is 5 times an odd number, of the form 5 ( 2 n + 1 ) 5(2n+1) . Now 5 ( 2 n + 1 ) = 10 n + 5 5(2n+1)=10n+5 has the last digit 0 + 5 = 5 0+5=5 .

Otto Bretscher - 5 years, 8 months ago

It is known ;)

Trần Phúc - 5 years, 8 months ago

Yeah We have same thinking.

Kushagra Sahni - 5 years, 9 months ago
Achille 'Gilles'
Sep 12, 2015

There is a 5 in the multiplications so last digit 0 or 5.

There is no even number so the last digit is odd so it is 5.

19 Helpful 0 Interesting 0 Brilliant 0 Confused
Chris White
Sep 14, 2015

No thought needed, 5 was involved and 0 wasn't an option thus only answer could be 5... took 2 seconds to answer with no math used.

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Sirajudheen Mp
Oct 19, 2015

since the product contain 5

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Didn't have even number in the question, just odd number can divided for 5. Easy to know consecutive even numbers can divide for 5 and have final digit is 5. So the answer is 5

0 Helpful 0 Interesting 0 Brilliant 0 Confused

odd number * 5 = 5

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Correct Logic.

Imran Zahid - 5 years, 8 months ago 
1
2
3
4
 
Not sure about this:
Each 'term' will only end in 1, 3, 5, 7, or 9
(1*3*5*7*9) mod (10) = 5
5*5*... mod 10 = 5

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Sadasiva Panicker
Sep 13, 2015

5xodd number =5

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Syed Baqir
Sep 13, 2015

Last Digit = ( 2000 + 15 ) 2015 → L a s t D i g i t i s 5 (2000+15)^{2015} \rightarrow \ Last \ Digit \ is \ 5 followed by 2000 zeros

0 Helpful 0 Interesting 0 Brilliant 0 Confused

ins't (2000+15 =2015) or have i done it wrong all my life!

Anirban Naskar - 5 years, 9 months ago

Log in to reply

You are correct but why you are asking this ??

Syed Baqir - 5 years, 9 months ago

look at the solution you posted mate.! then look at my previous comment. i think i can expect that you can understand it yourself!

Anirban Naskar - 5 years, 8 months ago

Log in to reply

Lol............. Thats typo ⌣ ¨ \huge \color{#D61F06}{ \ddot \smile }

Syed Baqir - 5 years, 8 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...