I don't know a solution without congurences, so if anyone could give a solution without one, I would be grateful.

Let the number of pearls be $N$ . Then, $N \equiv 1 \pmod 9 , N \equiv 5 \pmod 7$ . Therefore $N = 9a + 1$ . Putting this in the second equation gives.

$N = 9a + 1 \equiv 5 \pmod7$

$2a \equiv 4 \pmod7$

$a \equiv 2 \pmod 7$

Therefore $a = 7b + 2$ .

$N = 9a + 1 = 9(7b + 2) + 1 = 63b + 18 + 1 = 63b + 19$

Therefore $N \equiv 19 \pmod{ 63}$ . Since $23 < N < 100 , N = 63 + 19 = 82.$ . Therefore number of pearls not found $= 82 - 23 = \boxed{59} \\$

Series Possibility ( For finding same value of output and must consider pearls quantity is less 100) 9 x11 +1 = 100 ; 7 x14 +5 = 103 Not yet matching & also values are higher then 100. Go next---------- 9 x10 +1 = 91; 7 x13 +5 = 96 Not yet matching. Go next--------- 9 x10 +1 = 91; 7 x13 +5 = 96 Not yet matching Go next-------- 9 x9 +1 = 82; 7 x11 +5 = 82 Matching (Further these series be reached in 44 but question is marked that pearls quantity fewer then 100 ) So very close value is 82. Therefore lost pearly quantity is (82-23) = 59.

I made a a C++ program for this, and this is giving me the right answer, i.e 59.

include <iostream>

using namespace std; void main() { int found = 23; int missing = 0; for (int x = 99; x >= 0; x--) { if (x % 9 == 1 && x % 7 == 5) { cout << (x - found) << endl; break; }//end if x }//end for x }//end main