If d is a positive integer, and f is the product of the first 30 positive integers, what is the value of d?
(1) 10^d is a factor of f
(2) d>6
OA is C
Can someone kindly explain whY?
Grazie mille
Integer question
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f is the product of first 30 numbers.
This product will have 7 zeros (Zero comes in the product when any number is multiplied by 10,20 and 30 and any even number is multiplied by 5, 15 and 25)
so we have
5 x 2 = 10
then the number 10
15 x 6 = 90
then the number 20
25 x 4 = 100
then the number 30
So altogether f has 7 zeros.
1) Says 10^d is a factor of f. 10 (d=1), 100 (d=2), 1000 (d=3), 10000 (d=4)...etc upto 10^7 can be a factor of f. Hence not sufficient.
2) Says d > 6. d can be any positive integer greater than 6.
Both combined, we have d = 7. Hence sufficient
This product will have 7 zeros (Zero comes in the product when any number is multiplied by 10,20 and 30 and any even number is multiplied by 5, 15 and 25)
so we have
5 x 2 = 10
then the number 10
15 x 6 = 90
then the number 20
25 x 4 = 100
then the number 30
So altogether f has 7 zeros.
1) Says 10^d is a factor of f. 10 (d=1), 100 (d=2), 1000 (d=3), 10000 (d=4)...etc upto 10^7 can be a factor of f. Hence not sufficient.
2) Says d > 6. d can be any positive integer greater than 6.
Both combined, we have d = 7. Hence sufficient
- ssmiles08
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f = 30!tuscan21 wrote:I still don't understand why this product will have 7 zeros. Could you please elaborate on how this happens?Alara533 wrote:f is the product of first 30 numbers.
This product will have 7 zeros
the numbers in 30! that produce a zero at the end are
2*5*10*12*15*20*22*25*30 = 594x10^7
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ssmiles08 wrote:f = 30!tuscan21 wrote:I still don't understand why this product will have 7 zeros. Could you please elaborate on how this happens?Alara533 wrote:f is the product of first 30 numbers.
This product will have 7 zeros
the numbers in 30! that produce a zero at the end are
2*5*10*12*15*20*22*25*30 = 594x10^7
Why "10" for instance and not "11"?
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- lunarpower
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here's what you do:
forget entirely about 10, 20, and 30, and ONLY THINK ABOUT PRIME FACTORIZATIONS.
(TAKEAWAY: this is the way to go in general - when you break something down into primes, you should not think in hybrid terms like this. instead, just translate everything into the language of primes.)
each PAIR OF A '5' AND A '2' in the prime factorization translates into a '10'.
there are seven 5's: one each from 5, 10, 15, 20, and 30, and two from 25.
there are waaaaaaayyyyy more than seven 2's.
therefore, 30! can accommodate as many as seven 10's before you run out of fives.
--
statement 2 is clearly insufficient.
statement 1, by itself, means that d can be anything from 1 to 7 inclusive.
together, d must be 7.
ans (c)
forget entirely about 10, 20, and 30, and ONLY THINK ABOUT PRIME FACTORIZATIONS.
(TAKEAWAY: this is the way to go in general - when you break something down into primes, you should not think in hybrid terms like this. instead, just translate everything into the language of primes.)
each PAIR OF A '5' AND A '2' in the prime factorization translates into a '10'.
there are seven 5's: one each from 5, 10, 15, 20, and 30, and two from 25.
there are waaaaaaayyyyy more than seven 2's.
therefore, 30! can accommodate as many as seven 10's before you run out of fives.
--
statement 2 is clearly insufficient.
statement 1, by itself, means that d can be anything from 1 to 7 inclusive.
together, d must be 7.
ans (c)
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Pueden hacerle preguntas a Ron en castellano
Potete chiedere domande a Ron in italiano
On peut poser des questions à Ron en français
Voit esittää kysymyksiä Ron:lle myös suomeksi
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Quand on se sent bien dans un vêtement, tout peut arriver. Un bon vêtement, c'est un passeport pour le bonheur.
Yves Saint-Laurent
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Learn more about ron