If a = 15! + 13, which of the following are factors of a?
I. 13
II. 14
III. 26
A. I only
B. I and II only
C. I and III only
D. I, II, and III only
E. None of the above
OA A
Source: Veritas Prep
If a = 15! + 13, which of the following are factors of a? I
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A
B
C
D
E
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MULTIPLE OF X + MULTIPLE OF X = MULTIPLE OF X.BTGmoderatorDC wrote:If a = 15! + 13, which of the following are factors of a?
I. 13
II. 14
III. 26
A. I only
B. I and II only
C. I and III only
D. I, II, and III only
E. None of the above
MULTIPLE OF X + NON-MULTIPLE OF X = NON-MULTIPLE OF X.
Since the prime-factorization of 15! includes 13, we get:
a = 15! + 13 = multiple of 13 + multiple of 13 = multiple of 13.
Since a is a multiple of 13, option I is a factor of a.
Eliminate any answer that does not include option I.
Eliminate E.
Since the prime-factorization of 15! includes 2*7=14, we get:
a = 15! + 13 = multiple of 14 + non-multiple of 14 = non-multiple of 14.
Since a is not a multiple of 14, option II is a NOT factor of a.
Eliminate any remaining answer that includes option II.
Eliminate B and D.
Since the prime-factorization of 15! includes 2*13=26, we get:
a = 15! + 13 = multiple of 26 + non-multiple of 26 = non-multiple of 26.
Since a is not a multiple of 26, option III is a NOT factor of a.
Eliminate any remaining answer that includes option III.
Eliminate C.
The correct answer is A.
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Here's a similar question to practice with: https://www.beatthegmat.com/divisibility-t111432.html
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Let m, n, and p be positive integers. If m is divisible by k and n is divisible by k, then m + n is divisible by k. On the other hand, if m is divisible by k and n is NOT divisible by k, then m + n is NOT divisible by k.BTGmoderatorDC wrote:If a = 15! + 13, which of the following are factors of a?
I. 13
II. 14
III. 26
A. I only
B. I and II only
C. I and III only
D. I, II, and III only
E. None of the above
OA A
Source: Veritas Prep
Keeping the above fact in mind, let's analyze each Roman numeral choice.
I. 13
Since 15! is divisible by 13 and 13 is divisible by 13, then 15! + 13 is divisible by 13. In other words, 13 is a factor of 15! + 13.
II. 14
Since 15! is divisible by 14 and 13 is NOT divisible by 14, then 15! + 13 is NOT divisible by 14. In other words, 14 is NOT a factor of 15! + 13.
III. 26
Since 15! is divisible by 26 (15! has 2 and 13 as factors) and 13 is NOT divisible by 26, then 15! + 13 is NOT divisible by 26. In other words, 26 is NOT a factor of 15! + 13. Alternatively, we can also notice that 15! + 13 is odd (since 15! is even and 13 is odd) and 26 is even; thus 26 cannot be a factor of 15! + 13 since 15! + 13 has no factors of 2.
Answer: A
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$$a=15!+13$$
$$=13\left(\left(15\cdot14\cdot12!\right)+1\right)$$
For first option 13 ;
$$a=13\left(\frac{\left(15\cdot14\cdot12!\right)+1}{13}\right)=Odd\ number$$
Hence 13 is a factor of a,
$$Option\ I\ is\ correct$$
For second option 14 ;
$$a=13\left(\frac{\left(15\cdot14\cdot12!\right)+1}{14}\right)=even\ +\ remainder$$
Option II is wrong because a=odd and 14 is even ; even can't factor out odd without remainder.
For third option 26 ;
$$a=13\left(\frac{\left(15\cdot14\cdot12!\right)+1}{26}\right)=even+remainder$$
Option III is also wrong because a=odd and 26=even
even can factor out odd without remainder.
$$answer\ is\ Option\ A;\ I\ only\ $$
$$=13\left(\left(15\cdot14\cdot12!\right)+1\right)$$
For first option 13 ;
$$a=13\left(\frac{\left(15\cdot14\cdot12!\right)+1}{13}\right)=Odd\ number$$
Hence 13 is a factor of a,
$$Option\ I\ is\ correct$$
For second option 14 ;
$$a=13\left(\frac{\left(15\cdot14\cdot12!\right)+1}{14}\right)=even\ +\ remainder$$
Option II is wrong because a=odd and 14 is even ; even can't factor out odd without remainder.
For third option 26 ;
$$a=13\left(\frac{\left(15\cdot14\cdot12!\right)+1}{26}\right)=even+remainder$$
Option III is also wrong because a=odd and 26=even
even can factor out odd without remainder.
$$answer\ is\ Option\ A;\ I\ only\ $$