Can 5 odd numbers be added to get 30?

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Answer 1 · 2015-09-10T20:50:34

No.

The sum of an odd number of odd numbers is odd.

Every odd number can be written as ##2i+1# for an integrer, #i#, so
For this question in particular, if we add:

#2i+1#
#2j+1#
#2k+1#
#2l+1#
#2m+1#

We get:

#2(i+j+k+l+m) +5#

# = 2((i+j+k+l+m+2) +1#

which is of the form #2n+1#, so it is an odd number and cannot be equal to #30#.

Answer 2 · 2015-09-10T21:06:18

In normal integer arithmetic - no.

In modular arithmetic - yes with any odd modulo > 30.

Normal arithmetic

Suppose your #5# odd numbers are:

#a_1 = 2k_1 + 1#
#a_2 = 2k_2 + 1#
#a_3 = 2k_3 + 1#
#a_4 = 2k_4 + 1#
#a_5 = 2k_5 + 1#

where #k_1, k_2, k_3, k_4, k_5 in ZZ#

Then:

#a_1+a_2+a_3+a_4+a_5#

#= 2(k_1+k_2+k_3+k_4+k_5+2) + 1#

which is odd.

#30# is even - not odd - so #a_1+a_2+a_3+a_4+a_5 != 30#

Arithmetic modulo #31#

Let #a_1 = a_2 = a_3 = a_4 = 13# and #a_5 = 9#

Then #a_1+a_2+a_3+a_4+a_5 = 61 = 30# modulo #31#

In fact in modular arithmetic modulo an odd base all numbers are both odd and even.