Divisibility rules

Why divisibility tests?

To find the factors of a big number such as 8560, long division by every small number is slow. For some divisors, a glance at the last digits is enough. Asking whether 8560 is divisible by 10 is the same as asking whether 10 is a factor of 8560.

Divisible by 10, 5 and 2: the last digit

• By 10: the multiples of 10 are 10, 20, 30, … and all end in 0. So 8560 is divisible by 10, but 125 is not.
• By 5: the multiples of 5 end in 0 or 5. The largest number less than 399 that is divisible by 5 is 395.
• By 2: the multiples of 2 end in 0, 2, 4, 6 or 8, so 682 and 8560 are divisible by 2. Between 399 and 411, the multiples of 2 are 400, 402, 404, 406, 408 and 410.

These work because every number is a multiple of 10 plus its last digit, and 10 is a multiple of 2 and of 5.

Divisible by 4: the last two digits

The last digit is not enough for 4: 12 and 22 both end in 2, but 12 is a multiple of 4 and 22 is not. Look at the last two digits instead. Between 330 and 340 the multiples of 4 are 332 and 336; between 1730 and 1740 they are 1732 and 1736; and between 2030 and 2040, 2032 and 2036. The endings repeat, because 100 is a multiple of 4, so the hundreds and thousands never change the remainder. So 8536 is divisible by 4, because 36 is.

Divisible by 8: the last three digits

The last two digits are not enough for 8: 200 is a multiple of 8, but 100 is not. Because 1000 is a multiple of 8, the last three digits decide. Between 120 and 140 the multiples of 8 are 128 and 136; between 1120 and 1140, 1128 and 1136; between 3120 and 3140, 3128 and 3136. So 8560 is divisible by 8, since 560 = 8 × 70. Changing its last two digits to 68 gives 8568, another multiple of 8.

Combining tests

If a number is divisible by 8, it is also divisible by 4 and 2. If it is divisible by both 2 and 5, it is divisible by 10. So to check 2, 4, 5, 8 and 10 together, it is enough to check 8 and 5. That is how Guna knew that 14560 is divisible by all five after checking just two. Of 572, 2352, 5600, 6000 and 77622160, only 5600, 6000 and 77622160 pass both tests.

Leap years

A year is a leap year if it is a multiple of 4, except years divisible by 100 but not by 400. So 2000 was a leap year but 1900 was not. From 2024 to 2099 there are 19 leap years: 2024, 2028, …, 2096.

Palindromes divisible by 4

A 4-digit palindrome divisible by 4 must be even, so its first digit (the same as its last) is even, and its last two digits must form a multiple of 4. The largest is 8888 and the smallest is 2112.

To see how prime factorisation also tests divisibility, read prime factorisation. For the textbook’s questions, see the Chapter 5 NCERT solutions; for the whole chapter, see the Chapter 5 notes.