Divisibility · Topic 4 of 4

🧩 Many Divisors at Once

1 exam question type, fully solved

When a number must be divisible by a big composite divisor (like 72 or 75), or by several numbers at once, never long-divide. Break the condition into smaller co-prime pieces and apply the simple rules you already know.

The co-prime factor trick

If where and share no common factor (they are co-prime), then a number is divisible by exactly when it is divisible by both and . Useful splits:

• · · · · · .

• The factors must be co-prime: does NOT work, because passing the 6-test twice is not the same as passing both the 4-test and the 9-test.

• “Divisible by a, b and c”: the number must be a multiple of their LCM. If a, b, c are co-prime, the LCM is simply their product — e.g. and .

• Repeated-block patterns: a 2-digit block repeated gives ; a 3-digit block repeated gives , and .

⚡Co-prime Split CheckerPick a big divisor — we split it into co-prime pieces

72 =8 × 9(co-prime pieces — no shared factor)

⚠️ Trap: the pieces must be CO-PRIME (no common factor). 36 = 6 × 6 does NOT work — 12 passes the 6-test but is not divisible by 36. Split 36 = 4 × 9 instead: the two tests then cover different prime powers (2² and 3²), so together they truly prove ÷36.

To test 72, split it into the co-prime factors — check the last three digits for 8 and the digit sum for 9.

Q1A number is divisible by 6 exactly when it is divisible by:

Q2To test divisibility by 36, the safe split into co-prime factors is:

Q3A 6-digit number like 358358 (a 3-digit block repeated) is always divisible by:

📚

Real exam questions — Many Divisors at Once

1 question types · 5 solved examples from real SSC papers

Break a big or composite divisor into co-prime factors and apply each simple rule; use the LCM when a number must satisfy several at once.

How to solve this type

Never long-divide by a big composite number. Split it into CO-PRIME factors and apply each simple rule:

36 = 4\times 9

72 = 8\times 9

75 = 3\times 25

88 = 8\times 11

. (The factors must be co-prime —

6\times 6

does not work for 36.) When a problem says "divisible by a, b and c", the number must be a multiple of their LCM. Learn the repeated-block tricks too:

abab = ab\times 101

and

abcabc = abc\times 1001

, and

1001 = 7\times 11\times 13

A six-digit number is formed by writing a three-digit number twice in a row (for example 247247). It is always divisible by which number: 19, 111, 101 or 1001?

A19B111C101D1001

The 7-digit number 678p37q is divisible by 75 and p is not a composite number. Find p and q.

Ap=3, q=5Bp=3, q=0Cp=5, q=5Dp=2, q=5

If the 10-digit number 897359y7x2 is divisible by 72, find 3x − y for the largest possible y.

A7B3C5D8

The number 608xy0 is divisible by both 3 and 11, where x and y are non-zero digits in the hundreds and tens places. Find x and y.

A8 and 5B6 and 5C5 and 6D5 and 8

Find R and M if the number 34R05030M6 is divisible by both 16 and 11.

A5 and 7B5 and 5C4 and 6D7 and 5