Aptitude - Numbers - Discussion

x = 13p + 11 and x = 17q + 9.

The general case is,
x = 221a + 128 which give x = 349 for the least value of a = 1.

Let's try to demonstrate that!

Difference d > 0 get p = q + d and q = p - d.

We will replace q (into the second equality of x) to get p as function of d:
13p + 11 = 17(p-d) + 9 => 4p = 17d + 2.

We see now that d is even, so d = 2k and p = q + 2k.

4p = 17 x 2k + 2.

After dividing the above equality by 2 we will remark that k has an odd value so we will replace it by 2a+1.

2p = 17k + 1.

So k = 2a + 1 and p = q + 2(2a+1).

2p = 17(2a+1) + 1 => p = 17a + 9.

Replacing p in x = 13p + 11 will lead to the expected result.

x = 13(17a+9) + 11 => x = 221a + 128.

13p + 11 = 17q + 9.

q = (2 + 13p)/17.

To find the least value of p for which q = (2 + 13p)/17 is a whole number; The divisor must completely divide the dividend.

( i.e. denominator must completely divide the numerator.)

So the possible dividends are 170, 340, 510, ....

With Dividend 170,
(2 + 13p) = 170------------> p = 12.9..... , i.e. a fraction.

With dividend 340,
(2 + 13p) = 340 ------------> p = 26 i.e. a whole number.

Let X be the number which we have to find.

From given data we make two equations.

13k+11=X ---> (1)
17m+9 =X ---> (2)
From eq (1).

i.e.
13k+11 = X.
Adding 2 on both sides to make 11 as 13.
13k+11+2=X+2,
13k+13=X+2,
13(k+1)=X+2,
k+1=(X+2)/13,

From the options, X=349 is the only option is exactly dividable by 13.
Like vice eq(2).
Add 8 on both sides to 17.
At last, we get:
m+1=(X+8)/17.
So, X=349 is exactly dividable by 17.

@All.

According to me, "The least value of p for which. " ---the question didn't ask for the least value that satisfies the conditions. The question implies a unique number: "What is THE number?", not "What might be one such number/what is the smallest such number?".

Since the conditions given does not have a unique solution, "Data inadequate" should be the correct answer.

We can also find;
X=13p+11.
We can check values frm option and put in x.
i.e 349=13p+11.
338=13p.

Then check divisibility of 13.
If it is divisible by13 then it is the right answer.
Because 11 is the remainder if we subtract from the no. We will get an answer which will be exactly divisible by 13.

9 and 26 are the no's From which if we use it in an equation it will be divisible by 17.

So In this case, we can also use 9 as it is the least one, so 128 could be the answer, but there is no option for 128 that's why they have taken 26 as the least one and obtained the answer as 349.

Correct me if I am wrong.

Check from the options,

A) When 339 is divided by 13 we get the remainder as 1 but are should be 11 so check with the next option.

B) When 349 is divided by 13 we get the remainder as 11 and when it is divided by 17 are is 9, it satisfies both the conditions given in the question hence this is the answer.

Here is a simpler way for all of you.

Let no. be N, then;

N = 13a+11. > N-11 =13a (eq. 1).
N = 17b+9. > N-9 =17b (eq. 2).

Thus eq 1 and 2 suggest that N-11 should be divisible by 13 and N-9 should be divisible by 17.
N=349 satisfies both conditions!

if x = 13a - 11.
13a = x-11.
a = x-11/13
The ans a)339 sub in x place.

a = 339-11/13 = 25.2=> Not divisible by 13.
...take ans b)349 sub x place
a=349-11/13=26
So, 349 is completely divisible by 13 then the option B is the correct answer.

26 is a number by which if we use in a equation of q it is completely divisible by 17.

q=2+13(26) /17=20 that's why we take p=26.

Similarly if we put q=20 in eq.
P=17(20) -2/13=26,

So in both way
X=13(26) +1=349.
Y=17(20) +9=349.