If you have a 4 digit number and multiply it by 4, the product will be the number reversed. (abcd x 4= dcba) What is the number?
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To solve this problem, we need to find a 4-digit number that, when multiplied by 4, gives a product that is the number reversed.
Let's proceed step by step:
Step 1: Represent the 4-digit number as abcd, where each digit represents a place value.
Step 2: Multiply the number by 4: abcd x 4 = 4abcd
Step 3: Now, let's write down the product, 4abcd, in reverse order: dcba
Step 4: Set up an equation using the digits:
4a = a (since there is no carrying involved from a multiplied by 4)
4b = d (from the units place)
4c = c (from the tens place)
4d = b (from the hundreds place)
Step 5: Simplify the equations and solve for the variables:
From 4a = a, we get: 3a = 0. Since a cannot be zero, there is no solution for this equation.
From 4b = d, we get: b = d/4. Since b and d are digits, their possible values are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. So, d is a multiple of 4, and b can take the values 0, 1, 2, or 3.
From 4c = c, we get: 3c = 0. Since c cannot be zero, there is no solution for this equation.
From 4d = b, we get: 3d = b. Again, d can take the values 0, 1, 2, or 3.
Step 6: Substituting the values for b and d, we have the following possibilities:
When b = 0 and d = 0, we have 40. However, multiplying 40 by 4 gives 160, not the required number in reverse order.
When b = 1 and d = 4, we have 41. Multiplying 41 by 4 gives 164, which matches the required number in reverse order.
Step 7: Therefore, the number that meets the given conditions is 41.
Note: This solution is unique, as no other number satisfies the given condition.