All the even numbers from 2 to 98 inclusive, except those ending in
0, are multiplied together. What is the rightmost digit, that is,
the units digit of the product?
(a) 0 (b) 2 (c) 4 (d) 6 (e) 8
"Math is the study of patterns"
our multipliers are:
2,4,6,8,12,14,16,18,22,24,....94,96,98
multiplying them starting with 2x4 we get answers that end in
8,8,4,8,2,2,6,2, and then the same sequence repeats.
You will also notice that after each multiplier that ends in 8, like 18 or our last one 98, the answer ends in either 6 or 4.
So all we have to do is decide if it's a 4 or 6
Here is my reasoning for that.
There are 48 even numbers up to the 98, but we have to exclude mulipliers ending in 0 such as 10,20,..
there are 10 of these, so we are down to 40 multipliers resulting in 39 answers
Our loop of last numbers has a length of 8
39รท8 = 4.875 wich is near the end of the loop of reapeating digits,
so it must be the 6
To find the rightmost digit, or the units digit, of the product of all the even numbers from 2 to 98, except those ending in 0, we need to multiply all the even numbers together and then look at the units digit of the resulting product.
First, let's list all the even numbers from 2 to 98, excluding those ending in 0:
2, 4, 6, 8, 12, 14, 16, ..., 92, 94, 96, 98
To find the product, we multiply all these numbers together:
2 * 4 * 6 * 8 * 12 * 14 * 16 * ... * 92 * 94 * 96 * 98
To determine the units digit of this product, we can simplify the multiplication by only considering the units digits of each number.
The units digit of 2 is 2.
The units digit of 4 is 4.
The units digit of 6 is 6.
The units digit of 8 is 8.
The units digit of 12 is 2.
The units digit of 14 is 4.
The units digit of 16 is 6.
...
The units digit of 92 is 2.
The units digit of 94 is 4.
The units digit of 96 is 6.
The units digit of 98 is 8.
Now let's find patterns in these units digits to simplify the multiplication further:
Notice that the units digits repeat in cycles of 4: 2, 4, 6, 8.
We have 24 numbers in the list (from 2 to 98, excluding those ending in 0), so we have exactly 6 cycles of 4.
Since any multiple of 4 has a units digit of 6 (4, 8, 12, 16, ...), we can conclude that each cycle of 4 contributes a units digit of 6 to the overall product.
Therefore, we have 6 cycles of 4, and each cycle contributes a units digit of 6. So the units digit of the product is 6 * 6 = 36.
The rightmost digit, or the units digit, of the product of the even numbers from 2 to 98, excluding those ending in 0, is 6.
Therefore, the correct answer is (d) 6.
To find the rightmost digit of the product, we need to multiply all the even numbers from 2 to 98, except those ending in 0.
Let's list down the numbers to be multiplied:
2, 4, 6, 8, 12, 14, ..., 96, 98
Since we are looking for the rightmost digit, we can focus on the units digit of each number:
2 has a units digit of 2.
4 has a units digit of 4.
6 has a units digit of 6.
8 has a units digit of 8.
12 has a units digit of 2.
14 has a units digit of 4.
...
96 has a units digit of 6.
98 has a units digit of 8.
Notice that the units digit repeats in a cycle: 2, 4, 6, 8.
Since there is an equal number of even numbers with each units digit, the product of all these numbers will end in the same units digit as well.
When multiplying any number with a units digit of 2, 4, 6, or 8, the units digit of the product will always be even.
Therefore, the rightmost digit, or the units digit, of the product will be even.
Therefore, the answer is (a) 0.