The units digit of (44^91)*(73^37) is:

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The units digit of (44^91)*(73^37) is:

(A) 2
(B) 4
(C) 6
(D) 8
(E) 0



OA A

Source: Magoosh

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BTGmoderatorDC wrote:
Tue Jul 14, 2020 7:30 pm
The units digit of (44^91)*(73^37) is:

(A) 2
(B) 4
(C) 6
(D) 8
(E) 0

OA A

Source: Magoosh
To know the units digit of (44^91)*(73^37), we must the units digit of (4^91)*(3^37).

Let's understand the power cycle of 4 and 3.

4:

• 4^1 = 4 => unit digit = 4;
• 4^2 = 16 => unit digit = 6;
• 4^3 = 64 => unit digit = 4;
• 4^4 = 216 => unit digit = 6;

You will note that if the exponent of 4 is odd, the unit digit is 4 and if the exponent is even, the unit digit is 6.

Thus, the unit digit if (44)^91 = 4.

3:

• 3^1 = 3 => unit digit = 3;
• 3^2 = 9 => unit digit = 9;
• 3^3 = 27 => unit digit = 7;
• 3^4 = 81 => unit digit = 1;

• 3^5 = 243 => unit digit = 3;

You will note that after every 4 consecutive exponents, the units digit of the exponent of 3 repeats. They are in order 3, 9, 7 and 1.

Let's find out the units digit of 3^37.

3^37 can be written as 3^(36 + 1) = 3^(4*9 + 1)

So, the units digit of 3^37 = the units digit of 3^1 = 3

Thus, the units digit of (44^91)*(73^37) = units digit of 4*3 = units digit of 12 = 2.

Correct answer: A

Hope this helps!

-Jay
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BTGmoderatorDC wrote:
Tue Jul 14, 2020 7:30 pm
The units digit of (44^91)*(73^37) is:

(A) 2
(B) 4
(C) 6
(D) 8
(E) 0



OA A

Solution:

Let’s first determine the units digit of 44^91. Since we only care about units digits, we can rewrite the expression as:

4^91

Now we can evaluate the pattern of the units digits of 4^n for positive integer values of n. That is, let’s look at the pattern of the units digits of powers of 4. When writing out the pattern, notice that we are ONLY concerned with the units digit of 4 raised to each power.

4^1 = 4

4^2 = 6

4^3 = 4

The pattern of the units digit of powers of 4 repeats every 2 exponents. The pattern is 4–6. In this pattern, all positive exponents that are odd will produce 4 as its units digit, and all positive exponents that are even will produce 6 as its units digit. Since 91 is odd, the units digit of 4^91 (which is equal to the units digit of 44^91) is 4.

Next, let’s determine the units digit of 73^37 using a similar approach. Since we only care about units digits, we can rewrite the expression as:

3^37

Now we can evaluate the pattern of the units digits of 3^n for positive integer values of n. That is, let’s look at the pattern of the units digits of powers of 3. When writing out the pattern, notice that we are ONLY concerned with the units digit of 3 raised to each power.

3^1 = 3

3^2 = 9

3^3 = 7

3^4 = 1

3^5 = 3

The pattern of the units digit of powers of 3 repeats every 4 exponents. The pattern is 3–9–7–1. In this pattern, all positive exponents that are multiples of 4 will produce 1 as its units digit. Thus:

3^26 has a units digit of 1, and so 3^37 has a units digit of 3.

Since the units digit of 44^91 is 4 and the units digit of 73^37 is 3, the units digit of (44^91)*(73^37) is equal to the units digit of 4 * 3 = 12, which is 2.

Answer: A

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