Without using a calculator, decide which of these numbers divide into 2^5 * 3^9 * 7^2 * 13^5 8 17^4 * 19 with no remainder. Explain how you know.
a) 2^4 * 3^8 * 7^2 * 13^4 * 17^4 *
b) 2^5 * 3^2 * 5 * 7^2* 17
c) 2^4 * 13^4 * 17^4 * 19^4
d) 2^2 * 3^5 * 7 * 13^3 * 17^2 * 19
all divisors must have the same list of primes, with powers the same or less.
(b) does not work, since the dividend does not include a 5.
so, what is your answer?
I don't really understand that, doesn't b have a five in there?
yes, but b is the divisor
For it to divide the big number above (the dividend), there must also be a 5 there.
Ohhhh! Thank you!
I still dont get it. Can you explain the answer a bit more detailed?
To find out which of these numbers divide into the given number 2^5 * 3^9 * 7^2 * 13^5 * 17^4 * 19, we need to compare the exponents of the prime numbers present in the given number with the exponents in the options provided.
Let's analyze each option:
a) 2^4 * 3^8 * 7^2 * 13^4 * 17^4: To divide, the exponents of each prime number in the option should be less than or equal to the corresponding exponent in the given number. In this case, the exponents for 2, 3, 7, 13, and 17 are all less than or equal to the exponents in the given number. Therefore, option a) is a divisor of the given number.
b) 2^5 * 3^2 * 5 * 7^2 * 17: Again, we compare the exponents. In this case, the exponent for 3 is less than the exponent in the given number, but the exponents for 2, 5, 7, and 17 are all greater than the exponents in the given number. Hence, option b) does not divide the given number.
c) 2^4 * 13^4 * 17^4 * 19^4: Comparing the exponents, we see that the exponents for 2, 13, 17, and 19 are all less than or equal to the exponents in the given number. Thus, option c) is also a divisor of the given number.
d) 2^2 * 3^5 * 7 * 13^3 * 17^2 * 19: For this option, the exponents for 2, 3, 7, 13, and 17 are all less than the exponents in the given number, while the exponent for 19 is greater. Therefore, option d) does not divide the given number.
Therefore, the correct options that divide the given number without any remainder are a) 2^4 * 3^8 * 7^2 * 13^4 * 17^4 and c) 2^4 * 13^4 * 17^4 * 19^4.