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# Find the unit's digit in the product 2467^153*341^72

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Math Expert
Joined: 02 Sep 2009
Posts: 59124
Find the unit's digit in the product 2467^153*341^72  [#permalink]

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26 Sep 2019, 20:55
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Difficulty:

15% (low)

Question Stats:

82% (00:59) correct 18% (01:01) wrong based on 89 sessions

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Competition Mode Question

What is the unit's digit in the product $$2467^{153} * 341^{72}$$?

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

Find the unit's digit in the product (2467)^153 * (341)^72

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

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Re: Find the unit's digit in the product 2467^153*341^72  [#permalink]

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26 Sep 2019, 21:34
1
1
Unit digit of $$2467^{153}∗341^{72}$$ is same as unit digit of $$7^{153}∗1^{72}$$
= $$7^{153}$$
= $$7^{4*38 + 1}$$ (Since cyclicity of 7 is 4)
= $$7^1$$
= $$7$$

IMO Option D

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Re: Find the unit's digit in the product 2467^153*341^72  [#permalink]

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26 Sep 2019, 21:35
1
2467^153 x 341^72
Taking each of the terms separately and computing the unit digits correspondingly, we get
341^72
but the unit digit of 341 is 1. all powers of 1 will result in 1, hence the unit digit of 341^72=1
2467^153
the unit digit of 2467 = 7. The unit digits of the powers of 7 are as follows:
7^1=7
7^2=9
7^3=3
7^4=1
7^5=7
7^6=9
so we can see that the cyclicity of the powers of 7 is 4
hence 153/4 =38*4 + 1
Therefore the unit digit lies at the 5th position or 5th power of 7
i.e. unit digit of 7^153 = unit digit of 7^5 = 7

hence unit digit of 2467^153 x 341^72 = 7x1 = 7.

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Re: Find the unit's digit in the product 2467^153*341^72  [#permalink]

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26 Sep 2019, 21:35
1
Ans D - 7

2467^153∗341^72

Units digit of the product is dependent on the units digits of 2467^153 and 341^72. The units digits of 1^72 would be 1 because 1's cyclicity is 1.
Units digit of 2467^153 is 7. The cyclicity of 7 is 4.
7^0 =1
7^1 =1
7^2 = 9
7^3 = 3
7^4 = 1
7^5 = 7

So, the multiplication of these two numbers would be 7*1 = 7
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Re: Find the unit's digit in the product 2467^153*341^72  [#permalink]

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26 Sep 2019, 21:55
1
153=4K+1

7^4k+1 = last digit is 7
341^72 = last digit is 1
so 7*1 =7

OA:D
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Re: Find the unit's digit in the product 2467^153*341^72  [#permalink]

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26 Sep 2019, 23:16
1
Since unit digit of 341 is 1 which would always result in unit digit of 1, on multiplying with $$2467^{153}$$ it would be equal to unit digit of $$7^{153}$$.
$$2467^{153} * 341^{72} = 2467^{153} * 1^{72}$$
 $$= 7^{153}$$
 $$= 7^{38*4 + 1}$$
 $$= 7^{38*4} * 7^1$$
 = 1 * 7
 = 7

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Re: Find the unit's digit in the product 2467^153*341^72  [#permalink]

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27 Sep 2019, 03:38
cyclicity of 7 ; 7,9,3,1
unit digit of
2467^153 = 7
and 341^72 = 1
so answer IMO D ; 7

What is the unit's digit in the product 2467^153∗341^72?

(A) 0
(B) 1
(C) 2
(D) 7
(E) 9
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Re: Find the unit's digit in the product 2467^153*341^72  [#permalink]

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27 Sep 2019, 03:45
Quote:
What is the unit's digit in the product $$2467^{153}∗341^{72}$$?

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

$$units:2467^{153}∗341^{72}=7^{153}*1^{72}=7^{153}$$

$$cycles[7]:(7,9,3,1)=4$$

$$7^{153}…\frac{153}{4}=remainder[1]=1st.of=(7,9,3,1)=7$$

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Re: Find the unit's digit in the product 2467^153*341^72  [#permalink]

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27 Sep 2019, 08:01
7^153* 1^72;
153/4 = 1 remainder (and 0 remainder for 72/4)
7 ^1 * 1 ^4 = 7 * 1 = 7.
Ans. D
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Re: Find the unit's digit in the product 2467^153*341^72  [#permalink]

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27 Sep 2019, 08:36
Cyclicity of 7 is 4 and that of 1 is 1

Hence, unit digit of 7^153 is 7 and 1^72 is 1

Unit digit of the expression is 7

D is correct
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Re: Find the unit's digit in the product 2467^153*341^72  [#permalink]

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27 Sep 2019, 09:05
A number with 7 in unit's digit to the powers has the cycle of numbers 7,9,3,1 respectively.
So, the sycle is 4.
153÷4=38+1.
So, 2467^153 has 7 in its unit's digit.
7×1=7
Option D

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Re: Find the unit's digit in the product 2467^153*341^72  [#permalink]

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27 Sep 2019, 12:14
Units first number 7 >>> Therefore we need to know the cicle of 7 >>> (7,4,1,7,4,1) >>> The cicle repeats each three numbers.

153/3= 51 >> remainder = 0 therefore the units digit of the first number is 1 since the last number of the cicle is 1 (No remainder so last number of the cicle)

Units digit second number is 1, therefore the units digit of this number to any power will be 1.

1*1=1
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Re: Find the unit's digit in the product 2467^153*341^72  [#permalink]

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28 Sep 2019, 06:10
What is the unit's digit in the product $$2467^{153}∗341^{72}$$??

$$...7^{153}$$

$$...7^{1}=...7$$
$$...7^{2}=...9$$
$$...7^{3}=...3$$
$$...7^{4}=...1$$
..................
$$...7^{5}=...7$$
$$...7^{6}=...9$$
$$...7^{7}=...3$$
$$...7^{8}=...1$$
It is repeated in every four terms. So,
$$...7^{149}=...7$$
$$...7^{150}=...9$$
$$...7^{151}=...3$$
$$...7^{152}$$=...1
...$$7^{153}$$=...7

$$341^{72}=...1^{72}$$=...1
Units digit of this number always ends with 1.

Units digit is 7*1=7

.................
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Re: Find the unit's digit in the product 2467^153*341^72  [#permalink]

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29 Sep 2019, 07:48

Solution

Given
• We are given an expression: 2467^153∗341^72

To find
• The units digit of 2467^153∗341^72

Approach and Working out
• The units digit of 2467^153∗341^72 is same as the units digit of 7^153∗1^72
Units digits of 7^153∗1^72 = Units digits of 7^153∗ Units digits of 1^72
• = Units digits of 7^153 * 1
Units digits of 7^153 = Units digits of 7^(4*38 +1) = Units digits of 7^1 = 7

Hence, Units digits of 7^153∗1^72 = 7 * 1 = 7

Thus, option D is the correct answer.

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Re: Find the unit's digit in the product 2467^153*341^72   [#permalink] 29 Sep 2019, 07:48
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