A, B, C and D are positive integers such that A/B = C/D. Is C divisibl

our question can be interpreted as \(\frac{A*D}{B} = 5*K\)? where K, A, D, B - integers
#1 doesn't solve it coz B can be equal to A thus reducing the 210 completely and we are left with D which is unknown, INSUFFICIENT
#2 actually makes it look abit different, our denominator consists of 7's and if our numerator is divisible by 5 then so is C, but otherwise (if A*D is not divisible by 5) C is not , INSUFFICIENT
#1 + #2: combination of these makes our denominator a product of 7's which can't prevent number \(\frac{210*Z*D}{7^x}\) (A = 210*Z) from being divisibe by 5 due to number 210 alone being divisible by 5 and Z and D being positive integers thus giving us an explicit answer to the question. SUFFICIENT

C is the answer

Given \(\frac {A}{B}\,=\,\frac {C}{D}\)
\(C\,=\,\frac{A*D}{B}\)

Statement (i):
A is divisible by 210
A \(=\,210\,*\,integer\); no info about either B or D
Not sufficient

Statement (ii):
B \(= 7^x\), where \(x\) is a positive integer; no info about either A or D
Not sufficient

From (i) and (ii)
C \(=\frac {30\,*\,7\,*\,integer\,*\,D}{7^x}\)
\(\,\,\,\,\,= 30\,*\,some\,\,integer\); C is divisible by 5

Answer C

VERITAS PREP OFFICIAL SOLUTION

Let’s discuss the solution till the point I assume you will be quite comfortable.

We need to find whether C is divisible by 5. So let’s separate the C out of the variables.

C = AD/B

Since C is an integer, AD will be divisible by B but what we don’t know is that after the division, is the quotient divisible by 5?

Statement 1: A is divisible by 210

We still have no idea what B is so this statement alone is not sufficient. Let’s take an example of how the value of B could change our answer. Assume A is 210.

If B is 3, AD/B will be divisible by 5.

If B is 10, AD/B may not be divisible by 5 (depending on the value of D).

Statement 2: B = 7^x, where x is a positive integer

We have no idea what A and D are hence this statement alone is not sufficient.

Using both together: Now, this is where the trick comes in. Using both statements together, we see that C = (210*a*D)/(7^x)

Now we can say for sure that C will be divisible by 5.

C=AD/B
If we have at least one 5 as prime factor of C after the division the we can answer the question

Statement 1:
We know that A has at least (2,3,5 and 7 as prime factors). But we do not know the value of B. INSUFFICIENT

Statement 2:
We know that B has only 7 as prime factor, but we know nothing about the other numbers. INSUFFICIENT

Statements 1 and 2: Because we have 2, 3, 5 and 7 as prime factors of A*D and 7 is the only prime factor os B, we know for sure that we have at least one "5" as prime factor of C. SUFFICIENT

Excellent Question here we just need to make sure that either A or D has one 5 and that B does not cancel that 5 => C

Bunuel VeritasKarishma
Could anyone explain me this question
On combining both statements if I take the value of X as 100 then in that case, C= A*D/B will not be Integer.....
the How can the C be divided by 5
For eg take A = 210, D=1 and B = 7^35, In this case C will not be divided by 5.

As per my knowledge if something is divisible by some integer the it should yield the remainder as 0

Using both statements together, A is divisible by 210 so A = 210m (where m is some positive integer)
B = 7^x where x is an integer.

So what will A/B look like?

\(\frac{A}{B} = \frac{210*m}{7*7*7* ...7*7}\)

\(\frac{A}{B} = \frac{7*2*3*5*m}{7*7*7* ...7*7}\)

\(\frac{A}{B} = \frac{2*3*5*m}{7*7*7* ... 7}\)

If m has factors of 7, some more 7s from the denominator could get cancelled but that is it. The lowest form of A/B will have at least 2, 3 and 5 in the numerator and MAY have some 7s in the denominator.

Since \(\frac{A}{B} = \frac{C}{D} = \frac{2*3*5*...}{7*7*... *7}\)
So C will have factors of 2, 3 and 5 too.

As for your example, if A = 210 and B = 7^35, D cannot be 1.

\(\frac{A}{B} = \frac{210}{7^{35}} = \frac{2*3*5}{7*7*... 7}\)

Now, if C = 2*3*5, D = 7*7*...*7
if C = 2*3*5*2, D = 2*7*7*...*7
and so on...

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