whats the mystery number?

Question

if i divide the number by 3 the remainder is 1

if i divide the number by 4 the remainder is 2

if i divide the number by 5 the remainder is 3

if i divide the number by 6 the remainder is 4

Answer 1

First, PLZ remember that even multiples of a number have 0 remainder.  For example:  5, 10, 15, 20, … are even multiples of 5 and have a remainder of 0. So, the numbers that have a remainder of 3 when divided by 5 are 8, 13, 18, 23, …   (keep that in mind).

 

So, If we made a list of the four groups and found the Least Common Number, we would have found your magic number.  I like clues – does this term sound familiar?

 

If we made such a list, we would want to start with multiples of 6 with remainder 4 because all of the conditions must be true and this list has the fewest numbers.  We would then just check remainders:

           

   *6:    10, 16,  22, 28,  34, 40,  46, 52, 58 , 64,  70

   R6:      4,   4,    4,    4,   4,    4,    4,    4,    4,   4,   4

   R5:     0,    1,    2,    3,   4,    0,    1,    2,    3,   4,   0

   R4:     2,    0,    2,    0,   2,    0,    2,    0,    2,   0,   2

   R3:     1,    1,    1,    1,   1,    1,    1,    1,    1,   1,   1

 

Now, if we haven’t yet seen some pattern, we might not want to be a mathematician or any kind of scientist that needs math or …   But, what’s the pattern??

 

It’s called Modulo arithmetic and it simply creates the remainder (now, experts, PLZ restrict to positive numbers).

 

What “mystery number” has the specified remainders (remainders of 4, 3, 2, and 1)?   Looks like 58 is the Lowest Common Number.  There will be more (and they might be 58+60 and 58+120 and 58+180 and …).

 

So, if we determined that the Least Common Multiple (a different term) of 3 (prime), 4 (2*2), 5 (prime), and 6 (2*3) was 3*2*2*5 = 60, could we just subtract 2 (because the remainder of the magic number has a remainder of 4 when divided by 6) and always get the right remainders for the other factors??  And why would we subtract 2 rather than add 4?

Answer 2

The remainder of 3 when dividing by 5 means that it has to end with 3 or 8.  But rhe remainder of 2 when dividing by 4 means it has to be an even number, so it has to end with 8, not 3.

 

Look at the pattern numbers ending in 8 as they are divided by 3: 

 

8 / 3 = 2, remainder 2

18 / 3 = 6, no remainder

28 / 3 =  9, remainder 1

38 / 3 = 12, remainder 2

 

And the pattern repeats every 30.  So the number has to be 28 plus some multiple of 30.

 

28 doesn't work because 28 / 4 = 7 with no remainder.

 

The next possible number is 58.

 

58 / 3 = 19, remainder 1 -- check!

58 / 4 = 14, remainder 2 -- check!

58 / 5 = 11, remainder 3 -- check!

58 / 6 = 9, remainder 4 -- check!

 

So the number is 58

 

But actually, there are more numbers that work,  58 is just the lowest positive number that works.  After that, keep adding 60 and those numbers will work too.  That's because 60 divides evenly by 3, 4, 5, and 6, so it doesn't change the remainder compared to 58.