If the integer t is divided by 7, the remainder is 2. If t is divided

Question

If the integer t is divided by 7, the remainder is 2. If t is divided by 5, the remainder is 3. If t is between 15 and 75, what is the sum of all possible values of t?

(A) 38 
(B) 65 
(C) 81
(D) 103 
(E) 135

sujoykrdatta · Answer

t = 7n + 2
When t is divided by 5, rem is 3
=> t-3 is divisible by 5
=> (7n+2)-3 is divisible by 5
=> 7n-1 is divisible by 5
n = 3 satisfies (least value)

=> Least value of t is 7×3 + 2 = 23
Other values of t will be obtained by adding the lcm of the divisors i.e. 35
So, we have: 23 and 23+35=58 as the two values between 15 and 75
=> Sum = 81

Ans C

MintLog · Answer

sujoykrdatta 
Other values of t will be obtained by adding the lcm of the divisors i.e. 35 
how did you find this?

Bunuel · Answer

If the integer t is divided by 7, the remainder is 2. If t is divided by 5, the remainder is 3. If t is between 15 and 75, what is the sum of all possible values of t?

(A) 38 
(B) 65 
(C) 81
(D) 103 
(E) 135

Theory:  A positive integer  a  divided by a positive integer  d , leaving a remainder of  r , can be expressed as  a = qd + r,  where  q  is the quotient and  0 ≤ r < d .

If t leaves a remainder of 2 when divided by 7, then t = 7p + 2. So possible values of t are:

2, 9, 16,  23 , 30, ...

If t leaves a remainder of 3 when divided by 5, then t = 5q + 3. So possible values of t are:

3, 8, 13, 18,  23 , ...

Now we can derive a general formula based on both conditions. In problems of this kind, the combined pattern has the form  t = dx + r , where  d  is the least common multiple of the two divisors and  r  is the first common value in the two patterns.

Here, the two divisors are 7 and 5, so  d = LCM(7, 5) = 35 , and the first common value is r = 23. Therefore:

t = 35x + 23

So possible values of t are:

23, 58, 93, ...

The values between 15 and 75 are:

23 and 58

Thus, their sum is:

23 + 58 = 81

Answer: C.

sujoykrdatta · Answer

Because 35 is divisible by 5 and 7 and hence won't leave any extra remainders even if it is added to the number

If the integer t is divided by 7, the remainder is 2. If t is divided

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If the integer t is divided by 7, the remainder is 2. If t is divided

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