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T O P I C    R E V I E W
Katy Posted - 09/25/2009 : 20:07:24
Find the smallest positive integer that gives a remainder of 2 when divided by either 3 or 7, and gives a remainder of 3 when divided by 5.any help is greatly appreciated thanks :)
4   L A T E S T    R E P L I E S    (Newest First)
Ultraglide Posted - 01/13/2010 : 17:00:10
Just a note to Mrspi, you used bad form when writing
3*1=3+2=5
etc.
Subhotosh Khan Posted - 09/28/2009 : 10:07:13
quote:
Originally posted by Katy

Find the smallest positive integer that gives a remainder of 2 when divided by either 3 or 7, and gives a remainder of 3 when divided by 5.any help is greatly appreciated thanks :)



the number X = 3m + 2 and X = 7n + 2

then

3m + 2 = 7n + 2

3m = 7m

m = 7 and n = 3 are the smallest solution for the equation above (since 3 and 7 are relatively prime)

then

X = 3*7 + 2 = 23
Katy Posted - 09/26/2009 : 16:20:33
Thank you soooo much!!!
Mrspi Posted - 09/25/2009 : 21:42:08
quote:
Originally posted by Katy

Find the smallest positive integer that gives a remainder of 2 when divided by either 3 or 7, and gives a remainder of 3 when divided by 5.any help is greatly appreciated thanks :)



If the integer you're looking for gives a remainder of 2 when it is divided by 3, then it must be 2 MORE than a multiple of 3...I'll make a multiplication table for the 3's, and add two to each product

3*1 = 3 + 2 = 5
3*2 = 6 + 2 = 8
3*3 = 9 + 2 = 11
3*4 = 12 + 2 = 14
3*5 = 15 + 2 = 17
3*6 = 18 + 2 = 20
3*7 = 21 + 2 = 23
3*8 = 24 + 2 = 26

Ok, if we need to go farther than that, we can do it later.

the integer you're looking for gives a remainder of 2 when it is divided by 7, also. So, I'll make a multiplication table for 7, and add 2 to each product.

7*1 = 7 + 2 = 9
7*2 = 14 + 2 = 16
7*3 = 21 + 2 = 23
7*4 = 28 + 2 = 30
7*5 = 35 + 2 = 37

HMMMMMM...I see a possible candidate!! 23 is in BOTH of those tables!!

Could 23 be the number we're looking for? Let's check the last part, which says that "our number" has a remainder of 3 when divided by 5.

We're looking at 23...

23 divided by 5 is 4 remainder 3

Looking good!! And since there are NO smaller positive integers in the tables for 3 and 7, 23 must be the smallest positive integer which has a remainder of 2 when divided by 3 or 7, and a remainder of 3 when divided by 5.

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