If n is the smallest multiple of 10 which has the remainder of 2 when

Question

If n is the smallest multiple of 10 which has the remainder of 2 when divided by 3, and the remainder of 3 when divided by 7, how many positive factors does n have?

A. 6
B. 8
C. 9
D. 10
E. 12

Find the smallest multiple of 10 which has remainder 2 when divided by 3, and remainder 3 when divided by 7.

LipiSinha · Answer

Since, n is the smallest multiple of 10 which when divided by 7 leaves a remainder 3, let us first plug in n as 80. 
80 when divided by 7 indeed leaves a remainder 3. 
It also leaves a remainder 2 when divided by 3.

The prime-factorization of 80 is 2^4 * 5^1

Therefore, the number of positive factors of n will be (4+1)(1+1) = 5*2 = 10

The answer will be D.

TarunKumar1234 · Answer

If n is the smallest multiple of 10, then n belongs to 10, 20, 30, 40....
which has the remainder of 2 when divided by 3, then n belongs to 20, 50, 80....
and the remainder of 3 when divided by 7, then n belongs to 80....

So, n= 80 = 2^4 *5^1

No. of factors = 5*2= 10, So I think D.

XSatishX · Answer

Which is the first number which when divided by 3 has a remainder 2 and is a multiple of 10. Well it is 20. So all other numbers can be obtained by adding 30 to 20.. Hence all the numbers that will satisfy this are 20,50,80,110 and so on.

Now which number is a multiple of 10 and when divided by 7 leaves a 3. It is 10 itself. Now all other numbers can be deduced by adding 70 to 10. So all the numbers that will satisfy this condition are 10,80,150 and so on.

So what is the first common number in both the lists. As we can see 80 is the number that is the first number common in the list. Well if you want to find the other numbers, add 210 to 80 and so on. We are not done yet as the question asks for the number of factors.

Now prime factorization of 80 = 2^4*5^1
Number of factors = (4+1)*(1+1) = 5*2 = 10

Answer:  10

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If n is the smallest multiple of 10 which has the remainder of 2 when

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If n is the smallest multiple of 10 which has the remainder of 2 when

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