What is the units digit of (3^{101})(7^{103})?

[Max@Math Revolution]

[Math Revolution GMAT math practice question]

What is the units digit of (3^{101})(7^{103})?

A. 1
B. 3
C. 5
D. 7
E. 9

[GMATGuruNY]

[Math Revolution GMAT math practice question]

What is the units digit of (3^{101})(7^{103})?

A. 1
B. 3
C. 5
D. 7
E. 9

When an integer is raised to consecutive powers, the resulting units digits repeat in a CYCLE.

3¹�¹:
3¹ --> units digit of 3.
3² --> units digit of 9. (Since the product of the preceding units digit and 3 = 3*3 = 9.)
3³ --> units digit of 7. (Since the product of the preceding units digit and 3 = 9*3 = 27.)
3� --> units digit of 1. (Since the product of the preceding units digit and 3 = 7*3 = 21.)
From here, the units digits will repeat in the same pattern: 3, 9, 7, 1.
The units digit repeat in a CYCLE OF 4.
Implication:
When an integer with a units digit of 3 is raised to a power that is a multiple of 4, the units digit will be 1.
Thus, 3¹�� has a units digit of 1.
From here, the cycle of units digits will repeat: 3, 9, 7, 1...
Thus, 3¹�¹ has a units digit of 3.

7¹�³:
7¹ --> units digit of 7.
7² --> units digit of 9. (Since the product of the preceding units digit and 7 = 7*7 = 49.)
7³ --> units digit of 3. (Since the product of the preceding units digit and 7 = 9*7 = 63.)
7� --> units digit of 1. (Since the product of the preceding units digit and 7 = 3*7 = 21.)
From here, the units digits will repeat in the same pattern: 7, 9, 3, 1.
The units digit repeat in a CYCLE OF 4.
Implication:
When an integer with a units digit of 7 is raised to a power that is a multiple of 4, the units digit will be 1.
Thus, 7¹�� has a units digit of 1.
From here, the cycle of units digits will repeat: 7, 9, 3, 1...
7¹�¹ --> units digit of 7.
7¹�² --> units digit of 9.
7¹�³ --> units digit of 3.

Result:
3¹�¹7¹�³ = (integer with a units digit of 3)(integer with a units digit of 3) = integer with a units digit of 9.

The correct answer is E.

[Max@Math Revolution]

=>

The units digit is the remainder when (3^101)(7^103) is divided by 10.

The remainders when powers of 3 are divided by 10 are
3^1: 3,
3^2: 9,
3^3: 7,
3^4: 1,
3^5: 3,
...
So, the units digits of 3^n have period 4:
They form the cycle 3 -> 9 -> 7 -> 1.
Thus, 3^n has the units digit of 3 if n has a remainder of 1 when it is divided by 4.
The remainder when 101 is divided by 4 is 1, so the units digit of 3^101 is 3.

The remainders when powers of 7 are divided by 10 are
7^1: 7,
7^2: 9,
7^3: 3,
7^4: 1,
7^5: 7,
...
So, the units digits of 7^n have period 4:
They form the cycle 7 -> 9 -> 3 -> 1.
Thus, 7^n has the units digit of 3 if n has a remainder of 3 when it is divided by 4.
The remainder when 103 is divided by 4 is 3, so the units digit of 7^103 is 3.

Thus, the units digit of (3^101)(7^103) is 3*3 = 9.

Therefore, the answer is E.
Answer: E

[Scott@TargetTestPrep]

[Math Revolution GMAT math practice question]

What is the units digit of (3^{101})(7^{103})?

A. 1
B. 3
C. 5
D. 7
E. 9

Let's start by evaluating 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 digits 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 a 1 as its units digit. Thus:

3^100 has a units digit of 1, so 3^101 has a units digit of 3.

Next, we can evaluate the pattern of the units digits of 7^n for positive integer values of n. That is, let's look at the pattern of the units digits of powers of 7. When writing out the pattern, notice that we are ONLY concerned with the units digit of 7 raised to each power.

7^1 = 7

7^2 = 9

7^3 = 3

7^4 = 1

7^5 = 7

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

7^104 has a units digit of 1, and thus 7^103 has a units digit of 3.

So the units digit of (3^{101})(7^{103} is 3 x 3 = 9.

Answer: E

[regor60]

[Math Revolution GMAT math practice question]

What is the units digit of (3^{101})(7^{103})?

A. 1
B. 3
C. 5
D. 7
E. 9

(3^101)(7^103) can be rewritten as (3^101)(7^101)(7^2) = (21^101)(7^2) = 49(21^101)

Any number with 1 in the units digit raised to a power will continue to have 1 in the units digit, by simple inspection.

This units digit of 1 then being multiplied by 49 will have 9, E in the units digit