Units Digits

Question 1

Finding Units Digits:

Answer 1

In problems where you have to find the units digits of a number, GMAT is testing whether you know patterns of multiplication. i.e. in what pattern the units digits in multiples of a number change.

E.g. patterns for units digits of 4 raised to power:
4, 6, 4, 6, 4, 6, etc.

Question 2

Units Digits in Two-Digit Numbers Raised to Power:

Answer 2

The units digits of a two-digit number raised to a power is equivalent to the units digit of the units digit of the two-digit number raised to that power.

E.g. Units digit of 17^3 = units digit of 7^3

Question 3

Units Digit in Multiplication Problems:

Answer 3

If GMAT gives you a multiplication equation and wants to know what the units digit of the final number will be after you have computed the multiplication, the units of the final number will be equivalent to the product of the units digits of the separate factors.

E.g. units digit of 36 x 44 x 23 = units digit of 6 x 4 x 3 = 72
So units digit is: 2

Question 4

Units digits of Powers of 4:

Answer 4

Pattern for units digits of powers of 4:

4, 6, 4, 6, 4, 6 …

Question 5

Units digits for Powers of 6:

Answer 5

All powers of 6 end in a 6 as the units digits.

Question 6

Units digits of Power of 7:

Answer 6

Pattern for units digits of powers of 7:

7, 9, 3, 1, 7, 9, 3, 1, …

Question 7

Using Place Values in Digits Problems:

Answer 7

In some GMAT problems you have to use variables to represent unknown digits to be able to see properties of numbers.

E.g. A and B are both two-digit numbers, and A>B. If A and B contain the same digits, but in reverse order, what integer must be a factor of (A-B).

Here you should see quickly that what they want you to do is put variables as place holders as digits in A and B and then find out the properties of A-B.

Let’s say the digits are x and y.

A = xy
B = yx

Now you can’t calculate A - B with just that. You have convert A and B into real expressions to be able to do the calculation.

Now the only way to find factors of A-B is to re-express A and B as sums:

REMEMBER because you can use that a lot: A is the sum of the tens digit multiplied by 10 and the units digit. So, 10x + y.
And B is the sum of the tens digit multiplied by 10 and the units digit. So, 10y + x

So, A - B = 10x + y - (10y + x) = 10x + y - 10y - x = 9x - 9 y =
9 (x-y)

That means 9 is a factor of (A-B).

Question 8

Last Digit Shortcut for Questions about Units Digits:

Answer 8

To find the units digit of a product or sum of integers, only pay attention to the units digits of the numbers you are dealing with because only they affect the units digit of the result.

E.g. What is units digit of 7^2 x 9^2 x 3^2

Find units digits of the separate numbers:
7x7 = 49
9x9 = 81
3x3 = 9

So, units digit of 7^2 x 9^2 x 3^2 = 9x1x9 = 81 is 1.

Question 9

Unknown Digits Problems:

Answer 9

E.g. AB
xCA
——–
DEBC

In the multiplication above, each letter stands for a different non-zero digit, with A x B