S is the set of positive integers which, when divided by 2, 4, 5 and 8, leave remainders 1, 3, 4 and 7 respectively. How many integers between 0 and 100 belong to S?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 2
Each remainder is one less than its divisor, so adding 1 to such a number makes it exactly divisible by 2, 4, 5 and 8. The number plus 1 is therefore a multiple of the LCM.
\(4=2^{2}\), \(8=2^{3}\), and 2 and 5 are prime
LCM \(=2^{3}\times5=40\)
Number \(+1=40, 80, 120, \ldots\), so number \(=39, 79, 119, \ldots\)
Between 0 and 100: 39 and 79
Check 39: \(39=2\times19+1=4\times9+3=5\times7+4=8\times4+7\)
So 2 integers between 0 and 100 belong to S.