Find The Unit's Digit in the Product 2467^153*341^72 GMAT Problem Solving

Question: What is the unit's digit in the product \(2467^{153}∗341^{72}\)?

(A) 0
(B) 1
(C) 2
(D) 7
(E) 9

Correct Answer: D
Solution and Explanation:

Approach Solution 1:
We have been given an expression, which is ; 2467^153*341^72, and we have to find the unit's digit of this expression.
The units digit of 2467^153*341^72 is same as the units digit of 7^153*341^72.

The Units digits of 7^153*1^72 will be equal to the product of Units digits of 7^153 and Units digits of 1^72.
Therefore, Units digits of 7^153*1^72 = Units digits of 7^153 * Units digits of 1^72

Now we'll have to find the individual unit's of all these terms;
So, Units digits of 7^153 *1

We can also, write 153 as (4*38 +1)

Therefore, Units digits of 7^153 = Units digits of 7^(4*38 +1)
Thus, Units digits of 7^(4*38 +1) = Units digits of 7^1 =7
Therefore, from the above calculations, We get, Units digits of 7^153*1^72= 7 *1=7
Hence, the unit's digit in the product 2467^153*341^72 is equal to 7.

Approach Solution 2:
The Unit's digit of the whole term; 2467^153∗341^72, is dependent on on the unit's digits of 2467^153 and unit's digit of 341^72.
Therefore, to find the Unit's digit of the product of these terms, we'll have to find their individual unit's digit.
For the first term, the Unit's digit of 2467^153 is 7.

Now, the unit digits of the different powers of 7 are as per the following:
7^0 =1
7^1 =1
7^2 = 9
7^3 = 3
7^4 = 1
7^5 = 7

So, we can clearly see that the cyclicity of the powers of 7 is equal to 4.
Hence, 153/4 =38*4 + 1

Similarly, calculating for the second term, the Units digit of 341^72 is 1.
We know the cyclicity of 1 is equal to 1.

As, Cyclicity to 1= 1
Thus, the unit's digits of 1^72 would be equal to 1.
So, multiplying the unit's digits of both these terms. The unit's digit in the product 2467^153*341^72, will be, 7*1= 7.

Hence, the unit's digit in the product 2467^153*341^72 is equal to 7.

Approach Solution 3:
As we can clearly see the given expression; 2467^153*341^72, is the product of two individual terms; 2467^153 and 341^72. That's why we'll have to take both these terms separately and use them to find the unit's digit of 2467^153*341^72.

Thus, taking each of the terms separately and and separately and computing their corresponding unit's digits.

Firstly doing this for the term 341^72:
In the term 341^72, the unit digit of 341 is 1, and we know that all powers of 1 will result in 1.

Hence, the unit digit of the term; 341^72 will be equal to 1.
Unit's digit of 341^72 = 1
Now, doing the same steps for the second term, which is; 2467^153.

In the term 2467^153, the unit digit of the term is equal to 7.
Unit digit of 2467= 7

Now, the unit digits of the different powers of 7 are as per the following:
7^1=7
7^2=9
7^3=3
7^4=1
7^5=7
7^6=9

Thus, we can clearly see that the cyclicity of the powers of 7 is 4.
Hence, 153 can be written as;
153/4 =38*4 + 1

Therefore, we find out that the unit digit lies at the 5th position or 5th power of 7.
(unit digit of 7^153 = unit digit of 7^5= 7)

Hence, the unit digit of the expression 2467^153 x 341^72, which is basically the product of two terms; 2467^153 and 341^72, will be= 7x1= 7.
Hence, the unit's digit of 2467^153 x 341^72= 7

“Find The Unit's Digit in the Product 2467^153*341^72 GMAT Problem Solving”- is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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Find The Unit's Digit in the Product 2467^153*341^72 GMAT Problem Solving

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