If n is an integer between 10 and 100, is the tens digit of n even?

Question

If n is an integer between 10 and 100, is the tens digit of n even?
 
(1) The remainder when n is divided by 4 is equal to the remainder when n is divided by 5.
 
(2) The only prime factor of n is 3.

Bunuel · Accepted Answer

If n is an integer between 10 and 100, is the tens digit of n even? 
 
(1) The remainder when n is divided by 4 is equal to the remainder when n is divided by 5.

First of all, since the remainder is less than the divisor then the remainder must be 0, 1, 2, or 3.

n = 4q + r;
n = 5p + r;

4q + r = 5p + r;
4q = 5p --> q/p = 5/4 --> q must be a multiple of 5 and p must be a multiple of 4.

Thus, n = 4q + r = 4(5k) + r = 20k + r = (a multiple of 20) + r --> n could be 20, 21, 22, 23, 40, 41, 42, 43, 60, 61, 62, 63, 80, 81, 82, or 83. In any case the tens digit of n is even. Sufficient.

(2) The only prime factor of n is 3 --> n = 3^m. Since n is between 10 and 100, then it can be 3^3 = 27 or 3^4 = 81. In any case the tens digit of n is even. Sufficient.

Answer: D.

Hope it's clear.

KrishnakumarKA1 · Answer

ST 1: LCM is 20. hence the number can be 20, 21, 22, 23,40, 41,42,43, 60,61, 62, 63, 80, 81, 82, 83. All the cases, the tens digit is even. ANSWER

St 2: the possibilities are 03,09,27,81. All the cases, the tens digit is even.

Option D

TestPrepUnlimited · Answer

Statement 1:

Let the remainder be r, then we have:  n = 4*i + r  and  n = 5*j + r , where i and j are positive integers.

If we focus on  n - r , it is equal to  4i  and  5j . Thus  n - r  is a multiple of 4 and a multiple of 5, so  n - r  is a multiple of 20.

Finally, we have  n = 20*k + r , the remainder r is a digit less than 4 so the tens digit of n must be a multiple of 2, hence even.

Statement 2:

n = 3^i  where i is positive, but in order to land between 10 and 100 we need n to be 27 or 81, which both happen to have even digits in the tens place so this is sufficient.

Ans: D

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If n is an integer between 10 and 100, is the tens digit of n even?

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If n is an integer between 10 and 100, is the tens digit of n even?

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