2.3 Divisibility by 4

Rule: The number formed by the last two digits is divisible by 4 (00 counts as divisible). Reasoning: 100 is divisible by 4, so every part of the number except the last two digits (which is a multiple of 100) is automatically divisible by 4.

Example 1: 3,17,528 → last two digits = 28, 28÷4 = 7 exactly → divisible by 4. Example 2: 45,678 → last two digits = 78, not divisible by 4 → number not divisible by 4. Example 3: 5,83,916 → last two digits = 16, 16÷4=4 exact → divisible by 4. This rule is especially fast for checking 5-6 digit numbers where full division would otherwise cost valuable exam seconds.

Example 4 (edge case — “00” ending): Is 7,58,300 divisible by 4? Last two digits = “00”. Since 0 ÷ 4 = 0 remainder 0, “00” always counts as divisible — a common trap where aspirants wrongly assume a rule “breaks” when the last digits are zero. 7,58,300 is divisible by 4 (check: 758300 ÷ 4 = 189575 exact ✓).

Example 5 (edge case — every even digit works): Find all digits x (0–9) for which the number … x4 (i.e., a number ending in “x4,” where x is the tens digit) is divisible by 4. We need 10x+4 divisible by 4. Since 10x ≡ 2x (mod 4), the condition becomes 2x+4≡04, i.e., 2x≡04, i.e., x must be even. So x 0,2,4,6,8 — five valid digits, not one. This is exactly the kind of “not every missing-digit question has a unique answer” trap that Section 6 discusses in depth.