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If a = 15! + 13, which of the following are factors of a? I

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If a = 15! + 13, which of the following are factors of a? I

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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

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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.
$$answer\ is\ Option\ A;\ I\ only\ $$

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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
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.

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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scott@targettestprep.com



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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 + MULTIPLE OF X = MULTIPLE OF X.
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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Brent

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