What is the unit place digit in the expansion of 7 73 ?

Finding the Unit Digit of 7 Raised to a Power To find the unit digit of a number raised to a large power, we look for the pattern of the unit digits of the base number raised to successive powers. Let's find the pattern of the unit digits for powers of 7: $7^1 = 7$. The unit digit is 7. $7^2 = 49$. The unit digit is 9. $7^3 = 343$. The unit digit is 3. $7^4 = 2401$. The unit digit is 1. $7^5 = 16807$. The unit digit is 7. We can see that the unit digits repeat in a cycle: 7, 9, 3, 1. The length of this cycle is 4. To find the unit digit of $7^{73}$, we need to determine where in this cycle the 73rd power falls. We can do this by dividing the exponent (73) by the length of the cycle (4) and looking at the remainder. Divide 73 by 4: $$ \frac{73}{4} $$ We perform the division: $$ 73 = 4 \times 18 + 1 $$ The quotient is 18, and the remainder is 1. The remainder tells us which position in the cycle the unit digit corresponds to: If the remainder is 1, the unit digit is the first in the cycle (which is 7). If the remainder is 2, the unit digit is the second in the cycle (which is 9). If the remainder is 3, the unit digit is the third in the cycle (which is 3). If the remainder is 0 (or 4, meaning it's the end of a full cycle), the unit digit is the fourth in the cycle (which is 1). Since the remainder when 73 is divided by 4 is 1, the unit digit of $7^{73}$ is the same as the unit digit of $7^1$, which is 7. Revision Table: Unit Digits of Powers of 7 Power of 7 Result Unit Digit Position in Cycle (Remainder when exponent / 4) $7^1$ 7 7 1 (73 mod 4 = 1) $7^2$ 49 9 2 $7^3$ 343 3 3 $7^4$ 2401 1 4 or 0 Additional Information: Cyclicity of Unit Digits The unit digits of powers of any integer repeat in a cycle. The length of this cycle depends on the unit digit of the base number. Numbers ending in 0, 1, 5, 6 have a cycle length of 1 (the unit digit is always the same). Numbers ending in 4 or 9 have a cycle length of 2 (e.g., $4^1=4, 4^2=16$ (6), $4^3=64$ (4)... cycle 4, 6). Numbers ending in 2, 3, 7, 8 have a cycle length of 4. This concept of cyclicity helps quickly determine the unit digit of large powers without calculating the full number.

Unit Digit of 7^73 Calculation & Pattern

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Question

CDS - 2019

Quantitative Aptitude

What is the unit place digit in the expansion of 7 ⁷³?

This question was previously asked in

CDS I 2019 Elementary Mathematics Previous Year Paper (03-Feb-2019)

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The correct answer is

Finding the Unit Digit of 7 Raised to a Power

To find the unit digit of a number raised to a large power, we look for the pattern of the unit digits of the base number raised to successive powers.

Let's find the pattern of the unit digits for powers of 7:

$7^1 = 7$. The unit digit is 7.
$7^2 = 49$. The unit digit is 9.
$7^3 = 343$. The unit digit is 3.
$7^4 = 2401$. The unit digit is 1.
$7^5 = 16807$. The unit digit is 7.

We can see that the unit digits repeat in a cycle: 7, 9, 3, 1. The length of this cycle is 4.

To find the unit digit of $7^{73}$, we need to determine where in this cycle the 73rd power falls. We can do this by dividing the exponent (73) by the length of the cycle (4) and looking at the remainder.

Divide 73 by 4:

$$ \frac{73}{4} $$

We perform the division:

$$ 73 = 4 \times 18 + 1 $$

The quotient is 18, and the remainder is 1.

The remainder tells us which position in the cycle the unit digit corresponds to:

If the remainder is 1, the unit digit is the first in the cycle (which is 7).
If the remainder is 2, the unit digit is the second in the cycle (which is 9).
If the remainder is 3, the unit digit is the third in the cycle (which is 3).
If the remainder is 0 (or 4, meaning it's the end of a full cycle), the unit digit is the fourth in the cycle (which is 1).

Since the remainder when 73 is divided by 4 is 1, the unit digit of $7^{73}$ is the same as the unit digit of $7^1$, which is 7.

Revision Table: Unit Digits of Powers of 7

Power of 7	Result	Unit Digit	Position in Cycle (Remainder when exponent / 4)
$7^1$	7	7	1 (73 mod 4 = 1)
$7^2$	49	9	2
$7^3$	343	3	3
$7^4$	2401	1	4 or 0

Additional Information: Cyclicity of Unit Digits

The unit digits of powers of any integer repeat in a cycle. The length of this cycle depends on the unit digit of the base number.

Numbers ending in 0, 1, 5, 6 have a cycle length of 1 (the unit digit is always the same).
Numbers ending in 4 or 9 have a cycle length of 2 (e.g., $4^1=4, 4^2=16$ (6), $4^3=64$ (4)... cycle 4, 6).
Numbers ending in 2, 3, 7, 8 have a cycle length of 4.

This concept of cyclicity helps quickly determine the unit digit of large powers without calculating the full number.

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