Technique explanation
For larger divisors near powers of ten, we use block osculation by chopping groups of digits. If a divisor d = 10^n * k ± 1, the n-digit osculator is k. For composite divisors, factor them into coprime parts (GCD = 1) and test each part independently. For example, to test 28, test 4 and 7. Standard divisibility rules for simple factors serve as the fallback.
Standard fallback: Direct long division by the large divisor or testing each coprime factor using standard rules.
Worked examples
Example 1: Is 8,358 divisible by 199?
1. 199 is 200 - 1. The two-digit positive osculator is 2.
2. Chop two digits: 83 | 58. Apply P=2: 83 + 58(2) = 83 + 116 = 199.
3. Check: 199 is divisible by 199.
Check: Verify by multiplication: 199 × 42 = 8,358.
Result: 8,358 is divisible by 199.
Example 2: Is 1,512 divisible by 28?
1. Factor 28 into coprime parts: 4 and 7.
2. Test 4: The last two digits (12) are divisible by 4. (4 × 3 = 12).
3. Test 7: Use Q=2. 151 - 2(2) = 147. 14 - 7(2) = 0. Zero is divisible by 7.
Check: Verify by multiplication: 28 × 54 = 1,512.
Result: 1,512 is divisible by 28.
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