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# The number of digits in the number (4^11)(5^25) = ?

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The number of digits in the number (4^11)(5^25) = ?  [#permalink]

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28 Oct 2019, 21:23
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45% (medium)

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57% (01:04) correct 43% (01:06) wrong based on 72 sessions

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The number of digits in the number $$(4^{11})(5^{25}) =$$ ?

(A) 22
(B) 23
(C) 24
(D) 25
(E) 26

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Re: The number of digits in the number (4^11)(5^25) = ?  [#permalink]

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28 Oct 2019, 21:40
1
4^11*5^25=2^22*5^22*125=10^22*5^3=125*10^22, no of digits=3+22=25

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Re: The number of digits in the number (4^11)(5^25) = ?  [#permalink]

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28 Oct 2019, 21:49
To find the number of digits in 4^11 * 5^25

4^11=2^22
hence 4^11*5^25=2^22 * 5^22 * 5^3
But 2^22 * 5^22 = (2*5)^22=10^22
Therefore 2^22 * 5^22 * 5^3 = 10^22 * 5^3
but 10^a = a+1 digits
So, 10^22 = 22+1 = 23 digits
and 5^a = a digits
so 5^3 = 3 digits
Total digits = 23+3 = 26

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Re: The number of digits in the number (4^11)(5^25) = ?  [#permalink]

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28 Oct 2019, 23:22
2
1
To get the number of digits in any number:

1. Check the number of equal powers of 2's and 5's (i.e, $$2^x$$ and $$5^x$$). The powers should be same.
2. The remaining prime factors (including additional powers of 2 or 5 left out) should be multiplied to get a number.
3. Add the number of 0's from (1) and number of digits of the number from (2) to get the desired result.

(4^11)(5^25) can be written as (2^22)(5^25).

Step 1: The common powers of 2 and 5 are (2^22) and (5^22). This gives us 22 zeros
Step 2: The remaining prime factors and its powers are (5^3) = 125 which is 3 digits.

Step 3: Add no. of digits from 1 and 2 - (22+3 = 25)

Therefore, the answer is D 25.
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Re: The number of digits in the number (4^11)(5^25) = ?  [#permalink]

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29 Oct 2019, 00:18
1
4^11 x 5^25
= 2^22 x 5^25 (arranging 2s and 5s.....)
= 10^22 x 5^3 (.....to get 10s)
= 125 x 10^22

10^22 has 23 digits and multiplying it by 125 will add 2 digits more.

The factors will not add more digits as we have already merged 2s and 5s to 10s. In such questions we need to take care if the multiplication will add more digits, for e.g. 3 x 3 will yield a single digit number only whereas 3 x 5 will yield a double digit number.

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Re: The number of digits in the number (4^11)(5^25) = ?  [#permalink]

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29 Oct 2019, 00:25
125 followed by 22 zeros: Total: 25 digits
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Re: The number of digits in the number (4^11)(5^25) = ?  [#permalink]

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29 Oct 2019, 00:34
1
The number of digits in the number $$(4^11)(5^25) = ?$$

(A) 22
(B) 23
(C) 24
(D) 25
(E) 26
$$(4^{11})(5^{25})$$ = $$((2^2)^{11})(5^{25})$$
$$(2^{22})(5^{25})$$
$$(2^{22})(5^{22})(5^3)$$
$$(10^{22})(5^3)$$
22 + 3 = 25 digits

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Re: The number of digits in the number (4^11)(5^25) = ?  [#permalink]

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29 Oct 2019, 00:38
1
$$4^{11}*5^{25}$$= $$2^{22}*5^{25}$$=

$$5^{3}*5^{22}*2^{22}$$= $$5^{3}*10^{22}$$

—> $$5^{3}$$= 125 is greater than $$10^{2}$$, but less than $$10^{3}$$.

—> well, $$125*10^{22}$$= approximately greater than $$10^{24}$$, but cannot be equal to $$10^{25}$$.

—> $$10^{24}$$ — the number of digits in the number is 25

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Re: The number of digits in the number (4^11)(5^25) = ?  [#permalink]

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29 Oct 2019, 04:44
1
(4^11) (5^25) can be broken into (4^11) * (5^22) * (5^3)

4 * (5^2) = 100, which means with 4^11 and 5^22 we get 100^11 and the remaining 5^3 = 125.
1st part: 100^11 or 10^22 (22 zeros)
2nd part: 125(3 digits)
So, the number is 125 * 10^22 and the answer is D) 25
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Re: The number of digits in the number (4^11)(5^25) = ?  [#permalink]

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29 Oct 2019, 06:36
1
The number of digits in the number $$(4^{11})(5^{25})=?$$

$$4^{11} = (2^2)^{11} = 2^{22}$$

This means there are 22 $$(5 * 2)$$ pairs, leave $$5^3$$

$$5^3 = 125$$, 3 digits plus the 22 digits in the 5*2 pair equates to 25 digits

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Re: The number of digits in the number (4^11)(5^25) = ?  [#permalink]

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29 Oct 2019, 06:51
1
4^11*5^25 = 2^22*5^25
= (2*5)^22*5^3
= 125*10^22

Number of digits = 3 + 22 = 25

IMO Option D

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Re: The number of digits in the number (4^11)(5^25) = ?  [#permalink]

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29 Oct 2019, 07:42
1
$$4^{11}*5^{25}$$

$$= 2^{22}*5^{25}$$

$$= 2^{22}*5^{22}*5^3$$

$$= 10^{22}*5^3$$

10^22 - Will have 22 digits
5^3 = 125 , Will have 3 digits

SO, We have 22 + 3 = 25 Digits, Answer must be (D) 25
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Re: The number of digits in the number (4^11)(5^25) = ?  [#permalink]

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29 Oct 2019, 07:48
Digit with highest power becomes the base number. Other digits with their respective powers are not going to mutate the base number length.

Max(11,25) + 1
= 26
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Re: The number of digits in the number (4^11)(5^25) = ?  [#permalink]

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29 Oct 2019, 08:08
1
= 2^22 *5^22 * 5^3
= 625*10^22
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Re: The number of digits in the number (4^11)(5^25) = ?  [#permalink]

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29 Oct 2019, 08:35
1
((4^11)(5^25)
can be solveed
4^11*5^11*5^13
20^11*5^13
2^11*10^11*5^13
10^11 * 10^11 * 5^2
10^22 * 5^2
so we have 1 ; 1 ; 0 ; 22 ; 2 ; 1 and 5; 1
total 1+22+1+1 = 25 digits
IMO D

The number of digits in the number ((4^11)(5^25)= ?

(A) 22
(B) 23
(C) 24
(D) 25
(E) 26
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Re: The number of digits in the number (4^11)(5^25) = ?  [#permalink]

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29 Oct 2019, 08:59
1
D 25 ---> 2^22*5^22*5^3 = 10^22* 5^3 = 125* 10^22 . So we have 22 zeros and thre digits in 125 . Total digits = 3 + 22 = 25.
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Re: The number of digits in the number (4^11)(5^25) = ?  [#permalink]

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29 Oct 2019, 10:58
1
The number of digits in the number (411)(525)= ?

(A) 22
(B) 23
(C) 24
(D) 25
(E) 26

211. 211. 525= (2^11 *5^11) * (2^11 *5^11)* 5^3= 10^11*10^11*125= 10^22*125

=25 DIGITS.

IMO D
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Re: The number of digits in the number (4^11)(5^25) = ?  [#permalink]

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29 Oct 2019, 14:33
1
2²² × 5²² × 5³=
10²² ×5³= 125×10²²
3+22=25
Option D

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Re: The number of digits in the number (4^11)(5^25) = ?  [#permalink]

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29 Oct 2019, 19:09
1

2^22 *5^22*5^3
=5^3*10^22
=125 and 22 zeros.
So, total 25 digits.
Re: The number of digits in the number (4^11)(5^25) = ?   [#permalink] 29 Oct 2019, 19:09
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