The sum of two odd numbers are divisible by 5

Question

The sum of two odd numbers are divisible by 5. The sum of cubes of these two numbers is M. What is the last digit of M?

A. 6
B. 5
C. 4
D. 0
E. 8

Bunuel · Answer

Probably the easiest way would be just to pick some numbers:

1 + 9 = 10 = multiple of 5.
1^3 + 9^3 = 1 + (something with the units digit of 9) = (something with the units digit of 0).

Answer: D.

Hope it's clear.

Kinshook · Answer

Given: The sum of two odd numbers are divisible by 5. 
Asked: The sum of cubes of these two numbers is M. What is the last digit of M?

Let the odd numbers be x & y 
x + y = 5k; k is even since sum of 2 odd numbers is even.

Let x = 3 & y = 7; x + y = 10
x^3 + y^3 = 3^3 + 7^3 = 27 + 343 = 370

IMO D

Kinshook · Answer

Given: The sum of two odd numbers are divisible by 5. The sum of cubes of these two numbers is M.

Asked: What is the last digit of M?

Let the 2 odd numbers be x & y 
x + y = 5k = 10n; where k is an even integer and n is an integer since sum of 2 odd numbers is even.
M = x^3 + y^3 = (x+y)(x^2 + y^2 - xy) = 10n(x^2 + y^2 - xy)
Since x^3 + y^3 is a multiple of 10, last digit of M = 0

IMO D

VermaPulkit · Answer

An easy way to understand will be this:

For the number to be divisible by 5 it has to end with 0 or 5

Also sum of 2 odd number is always even. There fore units digit of sum will always be 0. Hence possible combinations of units digits are (order doesn’t matter as it’s addition)

1,9
3,7
5,5

Cubes of these combination will always gives 0 at units digit

Cube combinations of units digit will be (order doesn’t matter since it’s addition)

1,9
7,3
5,5

Hence sum of cube of numbers will always have units digit 0.

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Bunuel · Answer

Actually since we are not told that the numbers must be different, we can choose 5 and 5 to make everything much easier.

5 + 5 = 10 = multiple of 5.
5^3 + 5^3 = 125 + 125 = 250..

Answer: D.

gabrielneel · Answer

If we use 2 + 3 to give us 5, and if 5 is a multiple of 5, (5/5) = 1, we get the ones digit as a 5.

2^3 + 3^3 = 8 + 27 = 35.

Can you explain where I went wrong?

Krunaal · Answer

You probably missed "  The sum of    two odd numbers    are divisible by 5. The sum of cubes of these two numbers is M. What is the last digit of M?"

Both numbers should be odd, so taking 2 is invalid.

The sum of two odd numbers are divisible by 5

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The sum of two odd numbers are divisible by 5

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