Remainder

[MBA.Aspirant]

x is a positive integer <100. When x is divided by 5 remainder is 4. When x is divided by 23 remainder is 7. What's the value of x?

What's the easy way to go about this?

[MBA.Aspirant]

The only thing I got is using the remainder formula:

4 = x-5q

7= x - 23q

but it doesn't lead to anything

[Mom4MBA]

5a + 4 = x
23b + 7 = x

equate:

5a + 4 = 23b + 7
23b - 5a = -3

b can take values 0,1,2,3,4 above 4 x will be > 100, which is not allowed

for b=0, a=3/5 not possible
for b=1, a=26/5 not possible
for b=2, a=49/5 not possible
for b=3, a=72/5 not possible
for b=4, a=95/5=19
so x=99

[Roy@MasterGmat]

Good solution. Here is another approach for those less algebra-inclined:

Assuming that's the entire question, then the "what's the value of x?" phrasing implies numbers in the answer choices (each answer choice is a possible value for x).

If that's the case, trying out the answer choices is probably a much faster approach here. Divide each of the answer choices by 5 and 23, and choose the first option that results in the required remainders. Most test-takers could manage that much faster than any algebraic solution.

That being said, the equations are a great technique for those situations in which using the answer choices isn't possible.

As an alternative solution, it's also possible to try out all of the numbers below 100 that leave a remainder of 7 when divided by 23 (numbers greater by 7 than multiples of 23). There aren't that many of them under 100, so this shouldn't take too long.

23+7= 30, remainder of 0 when divided by 5

46+7= 53, remainder of 3 when divided by 5

69+7= 76, remainder of 1 when divided by 5

92+7= 99, remainder of 5 when divided by 5, and therefore the correct answer.

[GMATGuruNY]

x is a positive integer <100. When x is divided by 5 remainder is 4. When x is divided by 23 remainder is 7. What's the value of x?

What's the easy way to go about this?

Make a list of values for x.

When positive integer x is divided by the divisor D, the remainder is R.

Given the information above, the smallest possible value of x will be the remainder R.
To determine the other possible values of x, just keep adding multiples of the divisor D.

When positive integer x is divided by 23, the remainder is 7.
Smallest x = 7.
Now add multiples of 23:
7,30,53,76,99.

Now look for a value that will leave a remainder of 4 when divided by 5.
In other words, the correct value will be 4 more than a multiple of 5.

Only x=99 works:
99/5 = 19 R4.

The correct answer is [spoiler]x=99[/spoiler].