V4

[oquiella]

If a and b are two integers such that a is even, b is odd and neither of them leaves a remainder of 1 when divided by 10, what is the units digit of the product ab?

The units digit of a3 is the same as the units digit of a.
The units digit of b4 is the same as the units digit of b.


What is the general for these types of problems? Please breakdown

[MartyMurray]

If a and b are two integers such that a is even, b is odd and neither of them leaves a remainder of 1 when divided by 10, what is the units digit of the product ab?

The units digit of a³ is the same as the units digit of a.
The units digit of b� is the same as the units digit of b.

The key thing to realize when doing these units digit questions is that when multiplying integers, the digits other than the units digits do not affect the units digit of the product.

Check this out. The units digit of 3 x 3 = 9 is the same as the units digit of 233 x 843 = 196419.

Only the units digits affected the product's units digit, which is 9 in both cases.

So getting back to the question here, we are given restrictions on the units digits of a and b, and then asked what the units digit of ab is.

While we know from the question that neither of the units digits of a and b is 1, because we don't get a remainder of 1 when we divide either of them by 10, we don't yet have enough information to determine what the units digit of ab is.

Statement 1 further restricts the set of possible values of a. From the question we know that a is even. So the units digit could be 0, 2, 4, 6, or 8.

Now lets look at what happens when we cube numbers that have those units digits. Only the units digit affects the units digit. So we can merely look at what happens when we cube those possible units digits.

0³ has a units digit of 0.
2³ has a units digit of 8.
4³ has a units digit of 4.
6³ has a units digit of 6.
8³ has a units digit of 2.

So given the restriction provided by Statement 1, we are left with three possibilities. 0, 4, and 6.

We could pair a units digits, 0, 4, or 6, with various b units digits to get various ab units digits. So Statement 1 is insufficient.

Statement 2 further restricts b. From the question we already know that b cannot have a units digit of 1. So if b is odd b could be 3, 5, 7, or 9.

Now to satisfy Statement 2 we need a b such that the units digit of b� is the same as the units digit of b.

Since only the units digits matter we can try each of the possibilities to see which of them is the same as that number to the 4th power.

The units digit of 3� is 1.
The units digit of 5� is 5.
The units digit of 7� is 1.
The units digit of 9� is 1.

Only 5 works.

Now we know that b has a units digit of 5.

We already know that a is even.

HMMMMM

Any time we multiply an integer with a units digit of 5 by an even number we get an integer that is a multiple of 10, meaning that integer has a units digit of 0.

So from the information in the question and the information provided by Statement 2 we can determine that the units digit of ab is 0.

Choose B.

[oquiella]

If a and b are two integers such that a is even, b is odd and neither of them leaves a remainder of 1 when divided by 10, what is the units digit of the product ab?

The units digit of a³ is the same as the units digit of a.
The units digit of b� is the same as the units digit of b.

The key thing to realize when doing these units digit questions is that when multiplying integers, the digits other than the units digits do not affect the units digit of the product.

Check this out. The units digit of 3 x 3 = 9 is the same as the units digit of 233 x 843 = 196419.

Only the units digits affected the product's units digit, which is 9 in both cases.

So getting back to the question here, we are given restrictions on the units digits of a and b, and then asked what the units digit of ab is.

While we know from the question that neither of the units digits of a and b is 1, because we don't get a remainder of 1 when we divide either of them by 10, we don't yet have enough information to determine what the units digit of ab is.

Statement 1 further restricts the set of possible values of a. From the question we know that a is even. So the units digit could be 0, 2, 4, 6, or 8.

Now lets look at what happens when we cube numbers that have those units digits. Only the units digit affects the units digit. So we can merely look at what happens when we cube those possible units digits.

0³ has a units digit of 0.
2³ has a units digit of 8.
4³ has a units digit of 4.
6³ has a units digit of 6.
8³ has a units digit of 2.

So given the restriction provided by Statement 1, we are left with three possibilities. 0, 4, and 6.

We could pair a units digits, 0, 4, or 6, with various b units digits to get various ab units digits. So Statement 1 is insufficient.

Statement 2 further restricts b. From the question we already know that b cannot have a units digit of 1. So if b is odd b could be 3, 5, 7, or 9.

Now to satisfy Statement 2 we need a b such that the units digit of b� is the same as the units digit of b.

Since only the units digits matter we can try each of the possibilities to see which of them is the same as that number to the 4th power.

The units digit of 3� is 1.
The units digit of 5� is 5.
The units digit of 7� is 1.
The units digit of 9� is 1.

Only 5 works.

Now we know that b has a units digit of 5.

We already know that a is even.

HMMMMM

Any time we multiply an integer with a units digit of 5 by an even number we get an integer that is a multiple of 10, meaning that integer has a units digit of 0.

So from the information in the question and the information provided by Statement 2 we can determine that the units digit of ab is 0.

Choose B.

Hi Marty, what confuses me is the term "remainder" a remainder is usually .1 with a whole number in front of it. In this case it means numbers under 10?

[MartyMurray]

Hi Marty, what confuses me is the term "remainder" a remainder is usually .1 with a whole number in front of it. In this case it means numbers under 10?

A remainder is an integer or whole number concept.

When you divide a positive integer by a positive integer, if the division does not work out evenly, the integer that represents the extra amount is the remainder.

For instance, if you divide 8 by 5, the remainder is 3. If you divide 24 by 10, the remainder is 4. If you divide 256 by 25, the remainder is 6.

If you divide 100 or 110 or 250 by 10 the remainder is 0. Why? Well for one thing the units digit is 0, and any digit to the left of the units digit represents a number that is a multiple of 10.

If you divide 251, 471, or 102,621 by 10, the remainder is 1.