How Many Times Does One Number Go Into Another?

The phrase “how many times can I divide a number by another?” can mean a few different things. Most often, it asks how many whole groups of one number fit into another. This is a quotient-and-remainder problem. It may instead ask for the ordinary quotient, or how many times you can repeatedly divide and still get an integer.

Counting whole groups

Suppose you want to know how many times a positive integer $b$ goes into a positive integer $a$. There are unique integers $q$ and $r$ such that

$$a=bq+r, \qquad 0\le r<b.$$

Here, $q$ is the number of complete groups of size $b$, and $r$ is what remains. Equivalently, the number of whole groups is the floor of the quotient:

$$q=\left\lfloor\frac{a}{b}\right\rfloor.$$

For example, to find how many times 3 goes into 17, divide 17 by 3:

$$17=3\cdot 5+2.$$

So 3 goes into 17 5 whole times, with 2 left over. The decimal quotient, $17/3=5.666\ldots$, answers a different version of the question: it gives the exact result of division, not a count of complete groups.

When does one number divide another exactly?

A positive integer $b$ divides a positive integer $a$ exactly when the remainder is zero. In symbols, this is written $b\mid a$. For instance,

$$24=6\cdot4+0,$$

so 6 divides 24, and 6 goes into 24 exactly 4 times. By contrast,

$$25=6\cdot4+1,$$

so 6 does not divide 25 exactly: it goes in 4 whole times with 1 left over.

You can check exact divisibility by calculating the remainder, or by checking whether $a/b$ is an integer. A decimal result that merely looks close to an integer is not enough to establish exact divisibility; for integer inputs, use exact arithmetic rather than rounded decimal approximations.

If you mean repeated division

Sometimes “how many times can I divide by $b$?” means: how many consecutive times can I divide $a$ by $b$ and keep getting an integer? For positive integers $a$ and $b>1$, this count is finite. For example, starting with 72 and dividing by 2 gives

$$72\to36\to18\to9.$$

You can divide by 2 exactly three times. The fourth division gives $9/2$, which is not an integer. Equivalently, $2^3$ divides 72, but $2^4$ does not. If $b=1$, repeated exact division can continue indefinitely.

This repeated-division count is not the same as asking how many times 2 goes into 72 in one division: in that sense, 2 goes into 72 exactly 36 times. State which meaning you intend to avoid confusion.

Ordinary division and special cases

If you just want to divide $a$ by $b$, calculate $a/b$. The result need not be a whole number: $17/3=5.666\ldots$. The divisor must not be zero, because division by zero is undefined.

For positive whole numbers, the whole-groups rule is straightforward. If negative numbers are involved, quotient-and-remainder conventions can differ, so specify the convention being used. In everyday arithmetic, “how many times does it go in?” usually refers to nonnegative whole groups and a nonnegative remainder.

Common mistakes

• Confusing the quotient with the whole-group count: $17/3$ is $5.666\ldots$, but only 5 complete groups of 3 fit into 17.
• Ignoring the remainder: A remainder of zero means exact divisibility; a nonzero remainder means it does not divide exactly.
• Mixing up the two numbers: In “how many times does 3 go into 17?”, 17 is the dividend and 3 is the divisor. The calculation is $17/3$, not $3/17$.
• Dividing by zero: No number of groups or ordinary quotient is defined for a zero divisor.

For a quick check, multiply the proposed quotient by the divisor and add the remainder. You should recover the original number, and for whole-number division the remainder must be smaller than the divisor.